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THE  THEORY   OF  ELECTRIC 
CABLES     AND     NETWORKS 


THE  THEORY  OF 
ELECTRIC  CABLES 
AND  NETWORKS 


By 

ALEXANDER    RUSSELL,    M.A.,   D.Sc. 

Member  of  the    Council  of  the  Physical  Society, 
Member  of  the  Institution  of  Electrical  Engineers. 


OF  TH 


UNIVERS 
%/FOR 


U  Y 
:' 


NEW    YORK 

D.    VAN    NOSTRAND    COMPANY 

23   MURRAY  AND   27  WARREN   STREETS 
1909 


REESE 


BUTLER  &  TANNER, 

THE  SELWOOD  PRINTING  WORKS, 

FROME,  AND  LONDON. 


PREFACE 

THERE  is  nothing  more  conducive  to  the  satisfactory  work- 
ing of  an  electric  supply  station  than  having  a  thoroughly 
trustworthy  and  economical  network  of  cables  connecting 
the  dynamos  with  the  lamps  and  motors  of  the  consumer. 
It  is  necessary  therefore  that  the  engineer  have  a  thorough 
knowledge  of  the  phenomena  connected  with  the  flow  of 
current  along  conductors  and  across  dielectrics.  He  must 
also  have  a  working  knowledge  of  the  dielectric  strengths 
of  insulating  materials  and  the  electric  stresses  to  which 
they  are  subjected  under  working  conditions.  In  addition, 
the  thermal  conductivity  of  the  dielectric  has  to  be  con- 
sidered and  its  effect  on  the  temperature  of  the  conductor. 

The  author  gives  some  information  on  these  points  in  this 
book.  His  experience  in  practical  testing,  and  with  the 
difficulties  which  sometimes  arise  in  interpreting  "  specifica- 
tions "  and  "  rules  and  regulations  "  has  convinced  him  that 
the  solutions  of  these  problems  are  of  practical  use  and 
ought  to  be  more  widely  known.  In  fact  many  of  the 
problems  discussed  were  originally  suggested  by  these  diffi- 
culties. 

Questions  in  connexion  with  the  electrostatic  capacity 
and  the  inductance  of  cables  have  not  been  considered,  as 
the  author  has  discussed  these  points  fully  in  his  Treatise 
on  the  Theory  of  Alternating  Currents.  He  has  also 
omitted  many  elementary  theoretical  considerations  as  the 

V 

196523 


vj  PREFACE 

reader  is  supposed  to  know  the  elements  of  the  theory  of 
electricity  and  electrical  engineering. 

In  Chapter  I,  the  fundamental  electrical  principles  are 
stated  and  a  description  is  given  of  the  various  gauges  in 
use  for  specifying  wires.  Conductivity  is  discussed  in 
Chapter  II,  and  special  attention  is  devoted  to  the  effect 
of  the  "  lay  "  on  the  weight  and  conductivity  of  stranded 
cables.  In  Chapter  III,  the  standard  methods  of  measuring 
insulativity  are  described. 

The  design  of  distributing  networks  is  explained  in 
Chapter  IV,  particular  stress  being  laid  on  "  feeding  centres  " 
and  on  the  importance  of  calculating  their  positions.  The 
theorems  given  in  this  Chapter  can  easily  be  expanded  so 
as  to  enable  satisfactory  solutions  to  be  obtained  for  the 
very  complex  problems  which  sometimes  arise  in  practice. 

In  Chapters  V,  VI,  and  VII  methods  of  measuring  the 
insulation  resistance  of  house  wiring  and  distributing  net- 
works are  given.  The  author  only  gives  those  methods 
which  he  has  found  useful  in  practice.  The  problem  of  the 
calculation  of  a  suitable  resistance  to  put  in  the  earth  con- 
nexion with  the  middle  wire  was  suggested  to  him  by  Mr. 
A.  P.  Trotter. 

The  dielectric  strength  of  materials  is  discussed  in 
Chapter  VIII.  Unfortunately  very  few  accurate  data  are 
yet  obtainable,  but  the  author  hopes  that  by  applying  the 
methods  he  suggests,  engineers  will  be  able  to  obtain  satis- 
factory physical  "  constants  "  for  dielectric  strengths.  An 
examination  of  many  published  results  will  show  that  the 
experimenters  have  neglected  elementary  theoretical  con- 
siderations which  must  be  taken  into  account  if  the  results 
obtained  are  to  be  of  any  value. 

In  Chapter  IX,  the  grading,  and  in  Chapter  X,  the 
heating,  of  cables  is  considered.  It  is  only  of  recent  years 


PREFACE  vii 

that  the  former  of  these  subjects  has  been  recognized  to  be 
of  practical  importance.  In  Chapter  XI,  the  very  interest- 
ing subject  of  electrical  safety  valves  is  considered,  but 
only  a  few  types  are  discussed,  as  it  is  probable  that  the 
standard  safety  device  has  not  yet  been  evolved. 

The  author  has  added  a  Chapter  on  lightning  conductors, 
in  which  he  has  made  extensive  use  of  the  classical  paper 
on  the  subject  read  to  the  Institution  of  Electrical  Engin- 
eers by  Sir  Oliver  Lodge  in  1889. 

He  has  to  thank  several  friends  for  the  kind  help  they 
have  given  him  by  making  suggestions  or  revising  proofs. 
In  particular  he  has  to  thank  Dr.  Chree,  F.R.S.,  for  much 
information  about  atmospheric  electricity  and  Mr.  G.  F.  C. 
Searle,  F.R.S.,  for  his  helpful  criticisms  of  Chapters  I  and 
II.  He  has  also  to  thank  Mr.  J.  N.  Alty,  A.I.E.E.,  for  his 
able  assistance  in  drawing  the  diagrams  and  reading  proofs 
and  his  old  pupil,  the  Hon.  E.  Fulke  French,  for  checking 
most  of  the  mathematical  formulae  given. 

A.  R. 

10,  RICHMOND  BRIDGE  MANSIONS, 

TWICKENHAM. 
August,   1908. 


CONTENTS 


PAGE 

PREFACE  ..........         v 

CHAPTER  I 
FUNDAMENTAL  PRINCIPLES 


CHAPTER  II 
CONDUCTIVITY  .          .          .          .          .          .          .          .          .19 

CHAPTER  III 
INSULATIVITY   .          .          .          .          .          .          .          .          .49 

CHAPTER  IV 
DISTRIBUTING  NETWORKS  .......       67 

CHAPTER  V 
INSULATION  RESISTANCE  OF  HOUSE  WIRING        ...        95 

CHAPTER  VI 
INSULATION  RESISTANCE  OF  NETWORKS     , «      ,   .          .          .     Ill 

CHAPTER  VII 

FAULTS  IN  NETWORKS        .          .  ,          ,          ,          f     139 


x  CONTENTS 

CHAPTER  VIII 

PAGE 

DIELECTRIC  STRENGTH        ....  .163 

CHAPTER  IX 
THE  GRADING  OF  CABLES  .          .          .          .          .          .187 

CHAPTER  X 
THE  HEATING  or  CABLES  .          .          .          .          .          .211 

CHAPTER  XI 
ELECTRICAL  SAFETY  VALVES     .          .  .     225 

CHAPTER  XII 
LIGHTNING  CONDUCTORS.          .  .243 

INDEX  .          .          .  205 


FUNDAMENTAL    PRINCIPLES 


CHAPTER    I 

Fundamental   Principles 

Isotropic  bodies — Ohm's  law — Joule's  law — Example — Resistances 
in  series — KirchhofP s  first  law — The  potential  of  the  common 
junction — Minimum  heating — Conductors  in  parallel — Kirch- 
hoff' s  second  law — Minimum  heating  of  a  loop  of  a  network — 
Volume  resistivity — Section  variable — Volume  resistivities  of 
metals — Conductance  and  conductivity — Circular  mil — Gauges 
— Table  of  gauges — References. 

IN  this  chapter  we  shall  first  give  a  resume  of  the  elemen- 
tary electric  principles  which  guide  the  electrician  both  in 
the  design  of  a  direct  current  network  for  the  distribution 
of  electric  energy  and  in  the  measurements  of  the  electric 
properties  of  conducting  and  insulating  materials.  We 
shall  also  give  an  account  of  the  various  wire  gauges 
used  in  practice. 

Isotropic  *n  discussmg  the  electric  properties  of  con- 
bodies  ductors  it  is  customary  to  assume  that  the  con- 
ductors are  homogeneous  in  substance,  and  that  the  re- 
sistance they  offer  to  the  flow  of  current  through  them  is 
the  same  in  all  directions.  It  has  to  be  remembered, 
however,  that  violent  mechanical  forces  like  those  used  in 
hammering,  rolling  and  wire  drawing,  produce  permanent 
deformation  of  the  substance  of  the  metal,  and  alter  by 
varying  amounts  its  electrical  properties  in  different  direc- 
tions. In  this  chapter  we  shall  assume  that  the  conductors 
and  insulators  are  isotropic,  that  is,  that  their  substances 


4  ELECTRIC  CABLES  AND  NETWORKS 

have  the  same  physical  properties  in  all  directions.  Care- 
fully annealed  copper  may  be  considered  to  be  practically 
isotropic  as  no  tests  we  can  apply  can  detect  any  difference 
in  its  physical  properties  in  different  directions.  Carefully 
annealed  glass  also  is  practically  isotropic.  As  all  physi- 
cists, however,  now  accept  the  theory  of  the  molecular 
structure  of  bodies,  we  must  admit  that  if  the  portion 
examined  were  so  small  that  it  contained  only  a  few 
molecules  of  the  substance,  the  tests  for  isotropy  would 
not  be  satisfied.  All  substances  are  in  fact  irregular 
when  the  dimensions  of  the  portion  examined  are  compar- 
able with  the  dimensions  of  a  molecule  of  the  substance. 

If  R  be  the  resistance  of  an  electric  circuit, 
Ohm's  law 

E  the  electromotive  force  round  it,  and  C  the 

current  flowing  in  it,  Ohm's  law  states  that — 

0=E/B       (1), 

where  the  current  is  measured  in  amperes,  the  electro- 
motive force  in  volts  and  the  resistance  in  ohms. 

In  general,  if  r  be  the  resistance  of  a  part  of  a  circuit 
containing  sources  producing  a  resultant  electromotive 
force  E,  and  if  Fi  and  V2  be  the  potentials  of  A  and  B  the 
ends  of  this  portion  of  the  circuit,  we  have 

Cr=Vi— V2  +  E          (2). 

In  this  equation  the  current  C  is  positive  when  it  flows 
from  A  to  B,  and  E  is  positive  when  it  tends  to  produce 
a  current  in  the  same  direction. 

Equation  (2)  shows  that  it  is  possible  to  have  the  poten- 
tials Fi  and  F2  at  two  points  on  one  member  of  a  net- 
work of  conductors  the  same,  and  yet  have  a  current 
E/r  flowing  from  one  point  to  the  other.  In  this  case 
the  local  electromotive  force  E  is  entirely  expended  in 
maintaining  the  current  C  flowing  through  the  resistance 
r.  It  is  obvious  that  we  may  short-circuit  the  two 


FUNDAMENTAL  PRINCIPLES  5 

points  without  affecting  in  the  least  the  working  of  any 
part  of  the  network.  We  can  also  have  a  current  flowing 
from  a  point  of  lower  potential  through  a  source  of  elec- 
tromotive force  to  a  point  at  a  higher  potential. 

Joule's  -^y  *ke  definition  of  electromotive  force  given 
law  in  treatises  on  electricity  it  follows  that  if  a  current 
of  C  amperes  flow  for  t  seconds  through  a  wire  having 
a  potential  difference  of  V  volts  between  its  terminals,  the 
work  done  will  be  VCt  joules.  If  all  this  work  be  expended 
in  heating  the  wire,  that  is,  if  no  mechanical  work,  such  as 
causing  an  armature  to  rotate,  and  no  chemical  work,  such 
as  charging  accumulators,  is  done,  we  have 

JH  =  VCt      (3), 

where  H  is  the  number  of  water  gramme  Centigrade  units 
of  heat  (calories)  developed,  and  J  is  the  number  of  joules 
in  a  calorie.  The  law  expressed  by  this  equation  is  called 
Joule's  law,  as  he  was  the  first  to  employ  it  to  determine 
the  mechanical  equivalent  of  heat.  The  value  of  J  is  very 
approximately  4-18  ergs  per  water  gramme  degree  Centi- 
grade, and  hence,  by  Ohm's  law,  we  can  write 

JH=C*rt=(V*/r)t (4), 

or  H  =  0-239  <72rZ=  0-239(7 2 /r)t  .  .      ..      (5), 

approximately. 

Since  the  work  done  in  t  seconds  is  VCt  joules,  when  V 
and  C  are  maintained  constant,  it  follows  that  the  rate  at 
which  work  is  being  done  is  VC  joules  per  second,  that  is 
VC  watts. 

As  an  illustration  of  the  application  of  (5),  we 
Example  ,  . 

shall  find  the  rise  of  temperature  per  second  in 

a  coil  formed  of  a  copper  wire  0*1  mm.  in  radius  and  1,000 
metres  long  when  placed  between  the  hundred  volt  mains, 
supposing  that  no  heat  is  lost  by  radiation  and  neglecting 
the  effect  of  the  rise  of  resistance  due  to  rise  of  temperature. 


6  ELECTRIC  CABLES   AND   NETWORKS 

At  16°  C.,the  resistance  of  this  wire  would  be  about  508 
ohms.  By  (5),  the  heat  which  is  generated  per  second  is 

0-239  (100)2  /508, 

that  is,  4-7  calories  nearly.  Assuming  the  specific  gravity 
of  copper  to  be  8-9,  the  mass  of  the  copper  wire  will 
be  8-9  X7r(0-01)2x  100,000,  that  is,  280  grammes  nearly. 
Hence  taking  the  specific  heat  of  copper  to  be  0-095  the 
rise  of  temperature  per  second  will,  on  the  given  assump- 
tions, be  4-7/(280  x  0-095),  that  is,  0-18°  C.  nearly. 

We  conclude,  therefore,  that,  when  the  coil  is  connected 
with  the  mains,  its  temperature  rises  initially  by  about 
0-18°  C.  per  second.  As  it  warms,  the  heat  lost  by  radia- 
tion gradually  increases  and  so  the  rate  at  which  the  tem- 
perature rises  gradually  diminishes  until  the  temperature 
attains  a  steady  value,  when  the  rate  at  which  heat  is  being 
lost  by  radiation  equals  the  rate  at  which  heat  is  being 
generated  in  the  wire. 

If  we  have  n  coils  of  resistance  rl5  r2,  .  •/rn  respec- 

Resist- 

ances  in      tively,    connected  in   series,    and  if    C  be   the 
series 

current  flowing  through  them,  we  have,  by  (2), 

Cri  =  Vi~V2,Cr2  =  V2-V^  ..  Crn  =  Vn-Vn+1  ..  (6), 
where  Vv  and  Vv+l  are  the  potentials  at  the  ends  of  the 
resistance  rp.  It  follows,  by  adding  equations  (6),  that 


We  see,  therefore,  that  r^+r2+      ..      +rn  is  the  resultant 
resistance  R  of  the  n  coils,  so  that 

B  =  ri+ra+     ..      +rn     ......     (7). 

Hence  the  resistance  of  n  coils,  in  series  equals  the  sum  of 
the  resistances. 

When  we  have  n  conductors   connected  with 
Kirch- 

hoff's  first     a  point  0  and  the  currents  in  them  have  attained 
law 

their  steady  values,  the  algebraical  sum  of  all 

the  currents  in  these  conductors  must  be  zero.     For  if  not, 


FUNDAMENTAL  PRINCIPLES  7 

the  quantity  of  electricity  at  0  would  continually  increase 

or  continually  diminish,  which  is  obviously  impossible.     In 

algebraical  symbols  we  may  express  Kirchhoff's  first   law 

as  follows  — 

C7i+C7a+    ..    +0W=0  ......      (8), 

or  simply,  %C=Q, 

a  current  Cp  being  positive  when  it  is  flowing  towards  the 

common  junction. 

The  poten-        Let  ^  ^e  tne  potential  of  the  common  junc- 

"cornL?6    tion  of  n  arms  of  a  network,  and  let  Fl5  F2,  .  . 

junction       y^  be  the  p0tentiais  of  the  other   ends  of  the 

arms,  then,  by  (2)  and  (8),  we  have 

(F1-F+^1)/r1+(F2-F+#2)/r2+    ..     +(Vn-V+En)/rn 

=0         ..........     (9), 

and  thus,  VS(l/r)  =S(  Vv+Ev)/rv. 

In  analogy  with  the  nomenclature  of  alternating  current 
theory,  conductors  having  a  common  junction  will  be  said 
to  be  star  connected. 

If  we  assume  that  the  potentials  at  the  ends 

Minimum     of  n  branches,  star  connected,  of  a  network  are 
•  heating 

constant,  and   that    their    resistances   and  the 

electromotive  forces  in  their  circuits  are  also  constant,  we 
see  almost  at  once,  by  the  differential  calculus,  that  (9) 
determines  the  value  of  F  which  makes 


a  minimum.  Hence  the  actual  value  of  the  potential  of 
the  common  junction  is  the  theoretical  value  which 
makes  the  heating  as  determined  by  Joule's  law  a  mini- 
mum. 

When  we  have  n  resistances  r1}  r2,      .  .     rn, 

in^puaUei     connected  in  parallel  between  two  mains  each 

of  negligible  resistance,  and  when  the  potential 

difference  F  between  the  mains  is  constant,  then  the  cur- 


8  ELECTRIC  CABLES   AND  NETWORKS 

rents  d,  C2,     .  .     Cn,  in  the  resistances  are  given  by 


and   therefore, 

7=0^=00*=  ..  =Cnrn  ..  ..  (10). 
Now  if  C  be  the  current  in  the  main,  we  have,  by 
KirchhofFs  first  law, 

C=01+0,+     ..     +Cn, 
and  hence,  by  (10), 

<7  =  F  (l/r1+l/r2+  ..  +!/>„)  ..  (11). 
If  R  be  the  value  of  the  single  resistance  which  when  placed 
between  the  mains  would  allow  a  current  C  to  flow,  we  have 


and  thus,  by  (11), 

l/U  =  l/r1+l/ra+  ...  +l/rn=Sl/r,  ..  (12). 
Hence  the  reciprocal  of  R  equals  the  sum  of  the  reciprocals 
of  the  resistances  of  the  coils. 

We  shall  call  R  the  equivalent  resistance. 
Since  7=  CR,  we  find  by  (10), 

Cl=C(R/r,)9Ca=C(R/r2)      ..      ..     (13). 

If  in  a  network  we  take  any  of  the  conductors 
Kirchhoff's 

second  which  form  a  closed  circuit,  the  algebraical 
sum  of  the  currents  multiplied  by  the  resist- 
ances of  these  conductors  equals  the  algebraical  sum  of  the 
electromotive  forces  round  the  closed  circuit.  This  theorem 
was  enunciated  by  Kirchhoff  and  is  known  as  his  second 
law.  It  follows  at  once  from  (2),  for 

SCr=S(V,-V,+1+li:,)=SE       ..     ..     (14), 
since  in  a  closed  circuit  2(FP  —  Vf+1)  =  Vi  —  F2+F2  —  F3 


If  in  a  network  we  choose  any  system  of  con- 
f      ductors  which  form  a  closed  circuit  and  if   the 
a  network      resultant  electromotive  force  round  this  circuit 
be  zero,  then,  for  all  values  of  the  currents  in 


FUNDAMENTAL  PRINCIPLES  9 

these  conductors  which  are  consistent  with  Kirchhoff's 
first  law,  the  values  which  make  the  heating  of  the  con- 
ductors a  minimum  satisfy  Kirchhoff's  second  law. 

Let  7*1,  r2,  .  .  be  the  resistances  of  the  various  con- 
ductors and  let  Ct,  C2,  .  .  be  the  currents  in  them.  Since 
C1  —  C2  gives  the  resultant  value  of  the  currents  flowing 
into  or  out  of  the  circuit  at  the  common  junction  of  r±  and 
r2,  and  as  this  value  is  to  be  the  same  whatever  hypothetical 
values  we  give  to  the  currents,  we  see  that  these  values 
must  be  C±  -{-x,  C2  -{-  x,  C3  +  x->  •  -  where  x  may  be 
positive  or  negative.  The  total  heating  W,  therefore,  is 
given  by 


But  by  Kirchhoff  's  second  law  2(7r  =  ^E  and  is  therefore 
zero.     Hence 


and  W,  therefore,  has  its  minimum  value  %C2r  when  x  is 
zero,  that  is,  when  the  values  of  the  currents  are  in  accord- 
ance with  Kirchhoff  's  second  law. 

The  volume  resistivity  p  of  a  conducting 
V°sSvitye"  substance  at  a  given  temperature  is  the  resist- 
ance offered  at  that  temperature  by  a  centi- 
metre cube  of  the  substance  to  a  flow  of  electricity  from 
one  face  to  the  opposite  face  of  the  cube,  the  lines  of  flow 
being  perpendicular  to  these  faces.  In  practice,  p  is  usually 
expressed  in  microhms  (millionths  of  an  ohm). 

Since,  by  Ohm's  law,  the  fall  of  potential  from  one  face 
to  the  other  of  the  cube  is  uniform  it  follows  that  the  resist- 
ance of  a  rectangular  prism  one  square  centimetre  in  cross 
section  and  the  nth  part  of  a  centimetre  long  is  p/n. 
We  see  that  the  resistance  of  a  rectangular  prism,  one 
square  cm.  in  cross  section  and  I  cms.  long  is  the  same 


10          ELECTRIC   CABLES   AND  NETWORKS 

as  that  of  nl  prisms  of  the  same  section  and  of  length 
l/n  arranged  in  series.  It  is  therefore  nl  x  (p/ri),  that 
is,  pi.  If  we  now  suppose  this  prism  divided  up  into 
m  parallel  prisms,  the  areas  of  the  ends  of  which  will  be 
the  mth  part  of  a  square  centimetre,  the  resistance  of 
each  of  these  elementary  prisms  will  be  mpL  If  we 
have  a  prism  of  length  I  and  cross  sectional  area  S,  we 
may  suppose  it  to  consist  of  mS  elementary  prisms 
arranged  in  parallel.  Its  resistance  would  therefore  be 
mpl/(mS),  that  is  pl/S.  If  R,  therefore,  be  the  resistance 
of  this  prism  in  microhms  we  have 

R=Pl/S (15). 

It  has  to  be  remembered  that  this  is  true  also  for  cylindrical 
conductors  of  any  section  since  a  cylinder  is  a  particular 
case  of  a  prism.  The  only  assumption  made  is  that  the 
current  flow  is  parallel  to  the  axis. 

Section  When  the  section  of  a  wire  varies  slightly, 

it  is  customary  in  calculating  its  resistance  to 
measure  the  cross  sectional  areas  at  equidistant  points  along 
the  wire,  and  to  substitute  the  mean  of  the  values  thus  found 
for  S  in  formula  (15).  To  see  the  nature  of  the  error  made  by 
this  assumption,  let  us  consider  the  resistance  R  of  a  series 
of  n  cylinders  each  of  length  l/n  and  of  cross  sectional  areas 
$1,  82,  •  •  Sn.  We  shall  suppose  that  the  cylinders  are 
joined  to  one  another  by  a  material  infinitely  thin,  having 
absolutely  no  resistivity,  and  spread  uniformly  over  their 
ends.  This  will  ensure  that  the  flow  of  the  current  at  every 
point  in  each  of  the  cylinders  is  parallel  to  its  axis  and  hence 
we  can  at  once  write  down  the  value  of  all  the  resistances. 
The  actual  value  of  the  resistance  of  the  whole  will  be 
greater  than  this,  for  the  stream  lines  of  current  will  be 
curved.  We  have,  by  (15), 


FUNDAMENTAL  PRINCIPLES 

The  formula  ordinarily  used  is 

h     ..      +Sn)/n} 


11 


Now,  by  algebra,  since  Si+S2  +  .  .  +Sn  is  greater  than 
n(SiS2  . .  Sn)l/n  and,  for  the  same  reason,  l/Si+l/S2  + 
. .  +  \/Sn  is  greater  than  n(8i8a  . .  Sn)~l/n,  we  have, 
therefore,  (^+^2+  ..  +Sn)(l/8i  +  l/Sa+  ..  +  1/SJ 
greater  than  n2,  and  hence  R  is  greater  than  R',  pro- 
vided that  Si,  S2,  . .  are  not  all  equal.  Since  the 
actual  value  of  the  resistance  of  the  n  rods  in  series  is  greater 
than  R,  it  is  a  fortiori  greater  than  R'.  Therefore  the 
value  of  the  resistance  of  a  wire  calculated  by  means 
of  (15)  by  making  the  customary  assumptions  is  too  small. 
Conversely  the  value  of  the  volume  resistivity,  calculated 
from  the  value  of  the  resistance  found  by  a  Wheatstone's 
bridge  by  aid  of  (15)  is  too  great.  If  the  wire  be  nearly 
uniform  in  cross  section  the  error  due  to  neglecting  the 
curvature  of  the  lines  of  flow  is  very  small. 

In  the  following  table  the  values  of  the  vol- 
Volume  re- 
sistivities of    ume  resistivities  of  pure  metals  at  0°  C.,  found 
metals 

by  J.   Dewar  and  J.  A.  Fleming  (Phil.  Mag. 

p.  299,  Sept.  1893)  are  given.  The  metals  were  in  all  cases 
soft  and  annealed. 


Metal 

•     P°u 

microhms 

Metal 

•p\ 
microhms 

Aluminium      .       .      . 
Cadmium  .... 
Copper 
Gold           '.      .... 

2-665 
10-02 
1-561 
2-197 

Nickel    .      .      .      ..  . 
Palladium   . 
Platinum 
Silver     /    '. 

12-32 
10-22 
10-92 
1-468 

Iron      

9-065 

Thallium      . 

17-63 

Lead. 

20-38 

Tin  *      *      .      .      f 

13-05 

IVEasrnesium             .      . 

4-355 

Zinc               .      .      * 

5-751 

12          ELECTRIC  CABLES   AND   NETWORKS 

The  conductance  K  of  a  conductor  is  mea- 
Conduct- 

ance  and      sured  by  the  current  flowing  in  the  conductor 
conductivity  .  .  . 

when  unit  potential  difference  is  applied  at  its 

terminals.     Hence,  by  Ohm's  law, 


and  so,  K  =  l/E. 

The  conductivity  K  of  the  substance  of  a  conductor  is 
measured  by  the  current  which  flows,  parallel  to  an  edge, 
through  a  unit  cube  of  the  substance,  when  unit  difference 
of  potential  is  maintained  between  the  two  faces  perpen- 
dicular to  the  edge.  Hence  it  readily  follows  that  the 
conductance  K  of  a  wire  of  conductivity  K,  length  I,  and 
cross  section  St  is  given  by 

K=*(8/l). 

As  K  and  K  are  simply  the  reciprocals  of  R  and  p,  it  is 
unnecessary  to  tabulate  their  values  as  the  values  of  the 
latter  quantities  for  various  wires  and  substances  are  given 
in  tables.  It  is  also  unnecessary  to  discuss  methods  of 
measuring  conductivity  separately  from  methods  of  mea- 
suring resistivity,  as  any  method  which  measures  the  one 
quantity  will  also  give  the  other. 

Circular  ^e  sna^  now  describe  how  the  dimensions  of 

mi1  the  conductors  used  in  practice  are  specified.  On 
the  Continent  of  Europe,  thin  wires  are  usually  specified 
in  terms  of  their  diameters  measured  in  millimetres.  In 
England  and  America  they  are  generally  specified  in  terms 
of  certain  gauges  or  in  terms  of  the  diameters  measured  in 
mils,  a  mil  being  the  thousandth  part  of  an  inch.  Cable 
manufacturers  call  the  area  of  a  circle  one  mil  in  diameter 
a  "  circular  mil."  If,  for  instance,  the  diameter  of  a  wire 
were  d  mils,  its  area  would  be  d2  circular  mils  or  0-7854 
d2/(l,000)2  square  inches  approximately,  since  the  value  of 
a  circular  mil  is  0-7854/(l,000)2  square  inches.  In  practice, 


FUNDAMENTAL  PRINCIPLES  13 

it  is  convenient  to  use  the  expression  circular  mil  as  it  is  a 
perfectly  definite  unit  and  by  its  use  we  avoid  the  necessity 
of  multiplying  the  square  of  the  diameter  by  7r/4,  i.e.  by 
0-7854. 

Various  gauges  are  used  for  the  measurement 
Gauges 

of  wires.     The  oldest  of  them  is  the  Birmingham 

Wire  Gauge  (B.W.G.).  In  this  gauge  the  thickest  wire 
which  is  tabulated  has  a  diameter  of  500  mils  and  is  denoted 
by  No.  00000.  The  thinnest  wire  has  a  diameter  of  4  mils 
and  is  called  No.  36.  In  England  this  gauge  has  been 
replaced  by  the  British  Legal  Standard  or  as  it  is  generally 
called  the  Standard  Wire  Gauge  (S.W.G.).  As  in  the 
B.W.G.,  the  thickest  wire  which  is  tabulated  has  a  dia- 
meter of  500  mils,  but  it  is  denoted  by  No.  0000000  or  7/0. 
The  thinnest  wire  is  No.  50  and  is  1  mil  in  diameter.  From 
the  tables  given  below  it  will  be  seen,  that  the  diameters  of 
the  wires  corresponding  to  the  various  numbers  do  not 
proceed  by  any  regular  law.  The  number  of  sizes  is 
ample  for  all  practical  purposes.  In  electric  lighting 
practice,  conductors  having  a  larger  sectional  area  than 
that  of  a  No.  14  S.W.G.  wire  are  stranded.  The  trolley 
wires  used  in  electric  traction  are  generally  No.  0,  3/0,  or 
4/0,  S.W.G.  and  have  diameters  of  324,  372,  and  400  mils 
respectively. 

In  America,  the  Brown  and  Sharpe  Gauge  (B.  &  S.)  is  in 
general  use.  It  is  the  only  gauge  that  has  been  calculated 
on  scientific  principles.  It  is  founded  on  the  Birmingham 
Wire  Gauge  but  the  diameters,  and  consequently  also  the 
areas,  of  the  cross  sections  of  the  wires  corresponding  to  the 
various  numbers  are  in  geometrical  progression.  The 
largest  wire  is  4/0  and  has  a  diameter  of  460  mils.  The 
smallest  is  No.  40  with  a  diameter  of  3-14  mils.  The 
diameter  of  every  wire  in  this  gauge  is  practically  double 


14 


ELECTRIC  CABLES   AND  NETWORKS 


that  of  the  sixth  consecutive  wire  succeeding  it  or  half  that 
of  the  sixth  consecutive  wire  preceding  it.  It  follows  that 
the  area  of  the  cross  section  of  every  wire  is  practically  half 
that  of  the  area  of  the  wire  which  is  three  above  it,  or  double 
that  of  the  wire  which  is  three  below  it.  For  instance  : — 


B.  &  S.  Gauge 

Diameter 
in  mils 

Area    in 
Circular  mils 

Mass  in  Ibs.  of 
1,000  yds.  Cu.  wire 

No.  0         ... 
No.  3  .... 

325 

229 

105,600 
52  630 

958 

477 

No.  6  .      .      .      . 

162       • 

26,250 

238 

It  will  also  be  seen  that  the  weight  of  a  yard  of  No.  n  wire 
will  be  half  that  of  a  yard  of  No.  (n — 3)  wire  and  double  that 
of  No.  (n-}-3)  wire. 

To  find  the  value  of  the  ratio  x  of  the  diameters  of 
consecutive  wires  in  this  gauge,  let  us  calculate  from  the 
diameters  of  No.  0  and  No.  10  wire  respectively.  These 
are  324-95  and  101-89  mils.  We  have,  therefore, 

324-95  =  101-89  xw 

and  hence,  10  log  x  =  log  324-95— log  101-89=0-5036849 
and  thus,  a;  =  1-123  very  nearly. 

The  diameter  of  4/0  wire  would  be  given  by  324-95 
(1-123)3,  that  is,  460-2  mils  nearly.  The  diameter  of  No. 
40  wire  would  be  324-95  (1-123)-40  which  equals  3-14  mils 
nearly. 

The  following  table  gives  the  diameters  of  the  wires  in  the 
Standard  Wire  Gauge,  the  Birmingham  Wire  Gauge,  and 
Brown  and  Sharpe's  Gauge.  The  masses  of  a  thousand 
yards  of  a  pure  copper  wire  of  the  various  sizes  are  given  in 
the  second  table  for  purposes  of  comparison. 


FUNDAMENTAL  PRINCIPLES  15 

THE  ENGLISH  AND  AMERICAN  GAUGES  (IN  MILS). 


No. 

S.W.G. 

B.W.G. 

B.  &  S. 

No. 

S.W.G. 

B.W.G. 

B.  &S. 

4/0 

400 

454 

460-2 

19 

40 

42 

35-9 

3/0 

372 

425 

409-6 

20 

36 

35 

32-0 

2/0 

348 

380 

364-8 

21 

32 

32 

28-5 

0 

324 

340 

324-9 

22 

28 

28 

25-3 

1 

300 

300 

289-3 

23 

24 

25 

22-6 

2 

276 

284 

257-6 

24 

22 

22 

20-1 

3 

252 

259 

229-4 

25 

20 

20 

17-9 

4 

232 

238 

204-3 

26 

18 

18 

15-9 

5 

212 

220 

181-9 

27 

16-4 

16 

14-2 

6 

192 

203 

162-0 

28 

14-8 

14 

12-6 

7 

176 

180 

144-3 

29 

13-6 

13 

11-3 

8 

160 

165 

128-5 

30 

12-4 

12 

10-0 

9 

144 

148 

114-4 

31 

11-6 

10 

8-9 

10 

128 

134 

101-9 

32 

10-8 

9 

7-9 

11 

116 

120 

90-7 

33 

10-0 

8 

7-1 

12 

104 

109 

80-8 

34 

9-2 

7 

6-3 

13 

92 

95 

72-0 

35 

8-4 

5 

5-6 

14 

80 

83 

64-1 

36 

7-6 

4 

5-0 

15 

72 

72 

57-1 

37 

6-8 

— 



16 

64 

65 

50-8 

38 

6-0 

— 

— 

17 

56 

58 

45-3 

39 

5-2 

— 

— 

18 

48 

49 

40-3 

40 

4-8 

MASS  OF  1,000  YARDS  OF  COPPER  WIRE  IN  POUNDS,  WHEN 
ITS  SPECIFIC  GRAVITY  is  8*90. 


S.W.G. 

No. 

Mass 
Lbs. 

S.W.G. 

No. 

Mass 
Lbs. 

S.W.G. 
No. 

Mass 
Lbs. 

S.W.G. 

No. 

Mass 
Lbs. 

4/0 

1452 

8 

232-3 

19 

14-52 

30 

1-395 

3/0 

1256 

9 

188-2 

20 

11-76 

31 

1-221 

2/0 

1099 

10 

148-7 

21 

9-293 

32 

1-058 

0 

952-7 

11 

122-1 

22 

7-115 

33 

0-9076 

1 

816-8 

12 

98-16 

23 

5-228 

34 

0-7682 

2 

691-3 

13 

76-82 

24 

4-393 

35 

0-6404 

3 

576-3  ! 

14 

58-08 

25 

3-630 

36 

0-5242 

4 

488-5 

15 

47-05 

26 

2-940 

37 

0-4196 

5 

407-9 

16 

37-17 

27 

2-440 

38 

0-3267 

6 

334-6 

17 

28-46 

28 

1-987 

39 

0-2454 

7 

281-1 

18 

20-91 

29 

1-679 

40 

0-2091 

16          ELECTRIC  CABLES  ASD  NETWORKS 

REFERENCES. 

J.  J.  Thomson,  Elements  of  the  Mathematical  Theory  of  Electricity 

and  Magnetism* 
J.  Dewar  and  J.  A.  Fleming.  "  The  Electrical  Resistivities  of  Metals 

and  Alloys  at  Temperatures  approaching  the  Absolute  Zero.** 

Phil.  Mag.  [5]  voL  xxxvi,  p.  271,  1893. 
F.  C-  Raphael,  "  The  Electrician  "  Wireman's  Pocket  Book. 


CONDUCTIVITY 


CHAPTER  II 
Conductivity 

The  elastic  constants  of  metal  wires — Hard  drawn  and  annealed 
copper— The  density  of  copper— The  standard  density— Mass 
resistivity — Resistance  temperature  formulae — Resistivity  tem- 
perature formulae— Numerical  example— Tinning— Measuring 
the  rise  of  temperature — Temperature  coefficients  of  metals — 
Stranded  cables— Effect  of  lay  on  the  mass  of  the  conductor 
— Effect  of  lay  on  the  resistance  of  the  conductor — Permissible 
current  in  cables — Resistance  of  cables — High  frequency 
alternating  currents— Data  for  calculations— References. 

THE   elasticity   of   an  isotropic   metal  depends 
constant^ ^f    on  two  qualities  of  the  metal,  its  resistance  to 

change  of  volume  and  its  resistance  to  change 
of  shape.  The  former  depends  on  the  compressibility  and 
the  latter  is  called  the  rigidity.  If  a  piece  of  metal 
recovers  its  original  volume  and  shape  exactly  when  the 
forces  applied  to  it  are  removed,  it  is  said  to  have  been 
strained  within  the  limits  of  perfect  elasticity.  Within 
these  limits  Hooke's  law — that  the  effects  produced  are  pro- 
portional to  the  applied  forces — is  true,  and  we  may  speak 
of  the  metal  as  being  perfectly  elastic.  It  is  of  importance 
that  engineers  should  know  the  elastic  constants  of  metals 
as  these  are  an  indication  of  their  suitability  for  certain 
purposes. 

If  when  a  body  is  subjected  to  any  forces  every  cubical 
portion  of  it  remains  a  cube,  although  its  volume  has 
altered,  this  strain  is  called  a  compression  when  the 
volume  has  diminished,  and  an  expansion  when  the 

19 


20          ELECTRIC  CABLES   AND  NETWORKS 

volume  has  increased.  The  bulk  modulus  k  of  the  sub- 
stance is  the  ratio  of  the  stress  to  the  strain.  If  a 
stress  of  dp  dynes  per  square  centimetre  uniformly  applied 
to  the  surface  of  a  body  of  volume  F  alter  its  volume  to 
V  —  dV,  the  compression  is  measured  by  dV/V,  and 
thus,  by  Hooke's  law 

k  =  stress/strain  =  —  dp/(dV/V)=  —V(dp/dV). 

In  the  preceding  case  we  have  considered  change  of 
volume  but  not  change  of  shape.  We  shall  now  consider 
change  of  shape  without  change  of  volume.  If  two  pairs  of 
the  bounding  faces  of  every  elementary  cubical  portion  of  a 
piece  of  metal  remain  squares  while  the  other  pair  of  faces 
become  parallelograms  having  angles  90°  -\-A  and  90°  —  A 
respectively,  the  piece  of  metal  is  said  to  be  sheared.  A 
simple  way  of  producing  a  shear  on  a  cube  is  to  apply 
tangential  stresses  of  p  dynes  per  square  centimetre  to  the 
four  faces  which  remain  squares,  the  stresses  on  opposite 
faces  being  oppositely  directed.  If  6  be  the  circular  mea- 
sure of  A,  0  will  equal  77^4/180,  and  the  rigidity  n  is  given  by 

n  —  stress/strain  =  p/0  dynes  per  square  centimetre. 

As  an  example,  let  us  suppose  that  a  tangential  stress 
of  108  dynes  per  square  centimetre  produces  a  shear  of 
7rxlO~4  radians  in  copper.  In  this  case 

n=(l/7r)xWl2=3-I8xlQ11  dynes  cm~2  approx. 

If  a  uniform  stress  of  T  dynes  per  square  centimetre  of 
cross  section  pull  out   a  uniform  rod  from  a  length  I  to  a 
length  l-\-\  cm.,  we  have  ;  — 
Young's  modulus  =E  =  stress/strain  =T/(\/l)  dynes  cm.~2. 

These  constants  are  not  independent  of  one  another. 
It  is  proved  in  treatises  on  elasticity  that  for  an  isotropic 
material 


and  therefore  E  cannot  be  greater  than  3n. 


CONDUCTIVITY 


21 


G.  F.  C.  Searle  (Phil.  Mag.  p.  199,  Feb.  1900,  or 
Experimental  Elasticity,  p.  113)  gives  the  following  values 
of  E  and  n  for  various  metals  and  alloys,  obtained  on  the 
assumption  that  the  wires  experimented  on  were  isotropic  :- — 


Metal 

E 
dynes  cm.  —  2 

n 
dynes  cm.—2 

"  Silver  "  Steel     

•98  X  1012 

7-87  x  1011 

Brass  (hard  drawn)    
Phosphor  Bronze  
Silver  (hard  drawn)    . 

•02 
•20 

•78 

3-72 
4-36 

2-82 

Copper  (hardened  by  stretching) 
Copper  (annealed) 

•24 

•29 

3-88 
4-02 

In  the  last  two  cases  E  is  slightly  greater  than  3r&.      This 
shows  that  the  wires  were  not  strictly  isotropic. 

As  a  rule  there  is  an  alteration  of  temperature  when  the 
volume  or  shape  of  a  body  is  altered.  Hence,  strictly 
speaking,  the  values  of  the  elastic  constants  are  indeter- 
minate unless  the  alteration,  if  any,  of  the  temperature  is 
specified.  It  is  customary  to  consider  two  cases,  namely, 
(1)  when  no  heat  is  allowed  to  enter  or  leave  the  body  during 
the  application  of  the  forces,  and  (2)  when  the  temperature 
of  the  solid  is  maintained  constant.  The  constants  ob- 
tained in  the  first  case  are  the  adiabatic  constants  k^,  E^ 
and  n^,  and  in  the  second  case  the  isothermal  constants 
kt)  Et  and  nt.  It  may  be  shown  by  aid  of  the  principles  of 
thermodynamics  that  in  the  case  of  copper  at  0°  C.,  k^  is 
about  3  per  cent,  greater  than  kt,  that  E$  is  about  0*3 
per  cent,  greater  than  Et,  and  that  n^  and  nt  have  practi- 
cally the  same  value.  For  most  purposes,  therefore,  we 
may  disregard  the  differences  between  Et,  nt  and  E^n^. 
Hard  drawn  *n  making  copper  wire,  a  drawplate  of  hard 

nealed1"       steel  pierced  with   several  holes   of  graduated 

copper        sizes  is  mounted  on  the  draw-bench.     The  wire 


22          ELECTRIC   CABLES   AND  NETWORKS 

is  drawn  in  succession  through  smaller  and  smaller  holes 
which  are  widest  where  the  wire  enters  and  taper  slightly  to 
where  it  leaves.  During  each  operation  it  is  wound  on  a 
reel  on  the  draw-bench.  After  this  process  the  wire  is 
hard-drawn.  It  is  usually  circular  in  section  having  been 
drawn  through  conical  holes.  By  properly  designing,  how- 
ever, the  holes  in  the  drawplate,  the  wire  can  be  drawn  so 
as  to  have  a  cross  section  of  any  desired  shape. 

Copper  is  annealed  by  heating  it  to  redness  and  cooling  it 
suddenly.  The  result  is  that  it  is  rendered  soft  and  malleable. 
In  electrical  work  it  is  customary  to  divide  copper  wires  into 
"  hard  drawn  "  and  "  annealed."  The  Engineering  Standards 
Committee  (England)  define  a  hard  drawn  copper  wire  as 
one  which  will  not  elongate  by  more  than  1  per  cent,  with- 
out fracture.  Such  wire  is  generally  used  for  overhead  con- 
ductors where  mechanical  strength  is  desirable,  as  its  break- 
ing stress  is  about  double  that  of  annealed  copper,  and  its 
conductivity  is  only  about  2  per  cent,  less  than  that  of 
the  soft  copper  wires  used  for  insulated  conductors  in 
electric  lighting. 

Benton's  ^'  -^'  Benton  nas  made  an  investigation 
experiments  (Physical  Review,  vol.  xiii,  p.  234,  1901)  on 
the  effect  of  "  drawing  "  on  the  elasticity  of  copper  wire. 
The  copper  wire  was  first  annealed  by  heating  it  elec- 
trically to  redness  and  cooling  it  suddenly.  The  effect 
of  successive  drawings  on  its  elastic  constants,  determined 
on  the  assumption  that  the  "drawn"  wire  was  isotropic, 
was  then  investigated.  The  wire  was  finally  annealed  and 
its  constants  were  again  found. 


CONDUCTIVITY 


23 


Treatment 

Wire 

Diameter 
in  cm. 

E 

dynes  cm.  —  2 

n 
dynes  cm.  —  2 

Annealed  
Drawn  once  
,  ,        twice 

A 

0-1504 
0-1391 

4-017  x  1011 
3-946 

,,        3  times    .... 

6' 
„         .... 

,,        9     ,,         .... 
Re-annealed  

0-1306 
0-1122 
0-0941 
0-0932 

1-387  xlO12 
1-390 
1-420 
1'190 

3-919 
3-876 
3-863 
4-322 

Annealed  

B 

0-1617 

1-282 

4-177 

Drawn  once  . 

0-1508 

1-321 

4-015 

,,        3  times    .... 
„        5     „        .... 

?»                    /            55                    .... 

„        9     „         .... 
Re  -annealed  .      .      .      .      . 

0-1366 
0-1230 
0-1140 
0-1001 
0-0986 

1-290 
1-326 
1-340 
1-332 
1-109 

3-961 
3-948 
3-945 
3-897 
4-305 

These  results  apparently  indicate  that  the  effect  of  suc- 
cessive drawings  is  to  increase  the  value  of  Young's  modulus 
and  to  diminish  the  value  of  the  rigidity,  but  since  E  is  in 
most  cases  greater  than  3n  the  assumption  of  isotropy  is 
not  strictly  permissible.  On  re-annealing  the  wire  the  value 
of  Young's  modulus  is  appreciably  smaller  than  its  initial 
value,  and  the  value  of  the  rigidity  is  appreciably  greater. 
Instead  of  considering  how  the  electrical 

The 

density  of      resistance  of  a  copper  wire  varies  with  the  length 
copper  . 

and   the  area   of   the   cross    section,  it  is  often 

convenient  in  practice  to  consider  how  it  varies  with  the 
length  and  the  mass  of  the  wire.  The  experiments  of 
Fitzpatrick  (B.  A.  Report,  1894)  prove  that  the  densities 
(mass  in  grammes  per  cubic  centimetre)  of  most  kinds  of 
commercial  copper  at  ordinary  temperatures  lie  between 
8-90  and  8-95. 

Since  the    coefficient    of  linear  expansion  of  copper  for 
rise  of  temperature  is  0-0000168,  we  have 

lt=  lo{  1-|-0-0000168^},  approximately, 


24         ELECTRIC  CABLES  AND  NETWORKS 

• 

where  10,  lt  are  the  lengths  of  a  copper  wire  at  0°  and  t°  C. 
The  volume  Vt  of  a  lump  of  copper  at  t°  is,  therefore,  given 

by 

Vt=V0{  1  +  0-0000168^3 

=  V0  { 1  +3  x  0-0000168Z }  approximately 

=  V0{1  +0-0000504Z}. 

Hence,  the  volume  of  a  given  mass  of  copper  increases  by 
about  the  half  of  1  per  cent,  for  a  rise  in  temperature  of 
100°  C.  Thus  the  density  of  copper  varies  appreci- 
ably with  the  temperature,  diminishing  by  about  0-005  per 
cent,  per  degree  as  the  temperature  rises. 

In  order  to   simplify  the  arithmetical  work 
The 

standard      necessary  in  making  calculations  and  to  assist 
density 

the    memory,    it   is   customary   in   England  to 

assume  that  copper  weighs  555  Ib.  per  cubic  foot  at  60°  F. 
(15-6°  C.).  Hence  the  volume  of  555  Ib.  of  copper  at 
0°  C.  is  taken  to  be  1, 728/(l +0-0000504  x  15-6),  that  is, 
1,726-6  cubic  inches.  Hence  at  0°  C.  1  Ib.  of  copper  has 
a  volume  of  3-111  cubic  inches  and  1  cubic  inch  has  a  mass 
of  0-3214  Ib.  at  the  same  temperature. 

Since  there  are  453-6  grammes  in  a  pound  and  16-39 
cubic  centimetres  in  a  cubic  inch,  it  will  be  seen  that  the 
standard  density  of  copper  at  15-6°  C.  is  taken  as  555  x 
453-6/(l,728x  16-39),  that  is,  8-890.  The  standard  density 
of  copper  at  0°  C.  is,  therefore,  taken  as  8-89  (1+0-0000504  x 
15-6),  that  is,  8-897  approximately. 

Mass  The  resistance  in  ohms  at  60°  F.  of    a  wire 

resistivity  one  metre  long  and  weighing  one  gramme  is 
called  the  mass  resistivity  of  the  metal  forming  the  wire. 
We  shall  denote  mass  resistivity  by  p '.  For  annealed  high 
conductivity  copper  the  standard  value  of  p'  assumed  in 
England  is  0-1508,  and  for  hard-drawn  high  conductivity 
copper  it  is  taken  to  be  0-1539. 


CONDUCTIVITY  25 

If  the  mass  of  a  wire  be  ra  grammes  and  its  length  be  L 
metres,  then,  if  //  be  the  mass  resistivity  and  R  the 
resistance  of  the  wire,  we  have 


To  prove  this,  notice  that  the  resistance  of  a  metre  of  the 
wire  weighing  m/L  grammes  would  be  p  '/(m/L),  i.e. 
p'L/m,  and  hence  the  resistance  of  L  metres  of  this  wire, 
having  a  total  weight  of  m  grammes  would  be  p'L2/m 
ohms. 

We  have  already  seen  that  if  p  be  the  volume  resistivity 
of  the  copper  (p.   10), 


where  R  is  measured  in  ohms,  p  in  microhms,  L  in  metres, 
and  S  in  square  centimetres. 

Hence,  if  F  be  the  volume  of  the  copper  and  d  its  density 
(8-89)  at  60°  F., 


Therefore,         10 *p'(L2/m)  =  p  (L*/m)  8-89  x  104, 

and  hence,  p'=P* 0-0889, 

and  p  =  //x  11-25,  at  60°  F. 

For  annealed  high  conductivity  copper,  for  instance, 
p  =  0-1508x1 1-25 

=  1-6965,  at  60°  F., 

and  for  hard  drawn  high  conductivity  copper, 
p  =  0-1539x11-25 

=  1-731,  at  60°  F. 

These  values  of  p  and  p/  are  taken  as  the  standard 
values  in  England,  and  are  generally  referred  to  as  Mat- 
thiessen's  Standards.  It  has  to  be  remembered,  however, 
that  they  are  only  consistent  with  one  another  when  the 
specific  gravity  of  the  copper  is  8-90.  But  the  values 
of  the  specific  gravities  met  with  in  practice  may  vary 


26          ELECTRIC  CABLES   AND  NETWORKS 

from  8-88  to  8-96.  It  will  be  seen,  therefore,  that  the  per- 
centage conductivity  of  a  sample  of  copper  wire  in  terms 
of  Matthiessen's  Standard  as  ordinarily  determined  may 
vary  by  as  much  as  the  half  of  1  per  cent,  according 
as  we  take  the  mass  resistivity  or  the  volume  resist- 
ivity standard.  The  conductivity  of  the  best  quality 
copper  used  in  practice  is  often  2  or  3  per  cent,  greater  than 
"  Matthiessen's  Standard." 

Resistance          ^^e    resis^ance    of    copper    wire    alters    con- 

tetaPrera~       siderably  with  temperature.     As  the  result  of 

formulae       an  extensive  series  of  experiments  Matthiessen 

(Phil.  Trans.,  Roy.  Soc.,   1862)  gave  the  following  formula 

for  the  connexion  between  the    conductance  Kt  of  a  wire 

at  t°  C.  and  its  conductance  K0  at  0°  C.— 

Kt=K0\l— 0-0038701£+0-000009009Z2]. 

Hence,  by  means  of  the  binomial  theorem,  it  is  easy 
to  show  that  the  formula  connecting  the  corresponding 
resistances  Rt  and  R0  is, 

^=JRo[l +0-00387^+0-0000060^  —  0 -0000000 12£3—  .  .  .]. 
As  the  error  introduced  by  the  assumption  that  the  relation 
between  Rt  and  R0  is  a  linear  one,  at  least  up  to  50°  C.,  is 
not  large,  this  assumption  is  generally  made  in  practice. 
We  assume,  therefore,  that 

Bt=R0(l+at) 

and  give  such  a  value  to  a  that  the  errors  due  to  this 
assumption  are  as  small  as  possible.  The  following  values 
of  a,  found  experimentally,  were  quoted  by  Dr.  Glazebrook 
in  the  Electrician,  vol.  59,  p.  65. 


NDUCT1V1TY 


27 


Observers 

Range  in 
Deg.  Cent. 

a 

Matthiessen  in  Phil.  Trans.  1862    .      .   j 

0-50 
0-100 

0-00412 
0-00422 

(These  mean    values  of    a  are  deduced 

from  Matthiessen'  s  formula.) 

0-18 

0-00424 

Dewar  and  Fleming,  Phil.  Mag.  1893.  - 

0-60 
0-100 

0-00427 
0-00428 

I 

0-200 

0-00426 

Swan  &  Rhodin,  Proc.  Roy.  Soc.  1894J 

13-90 

0-00408 
0-00416 

Fitzpatrick,  B.  A.  Report,  1894.     . 

— 

0-00405 

For  temperatures  between  0°  and  50°  C.  the  value  cus- 
tomarily taken  for  a  in  England  in  1908  is  0-00428.  In 
America  and  Germany  0-0042  is  taken  as  the  standard 
value.  It  is  probable  that  a  varies  appreciably  with  the 
kind  of  copper  used  and  the  method  of  treatment  to 
which  it  has  been  subjected,  but  sufficient  data  on  this 
point  have  not  yet  been  obtained.  We  shall  take  0-0042 
as  the  standard  value  for  the  temperature  coefficient  — 
for  temperatures  from  0°  to  50°  C.  The  formula  is 
therefore  — 


Resistivity 

temperature 

formulae 


Assuming  that  the  formula 

RJM*  =  1  +  at 

gives  the  value  of  Rt  accurately,  we  shall  find 
the  temperature  coefficients  for  the  volume  resistivity  and 
for  the  mass  resistivity.  For  the  volume  resistivity,  we  have 

Pt=Et(St/lt)  and  Po=  R0(S0/10), 
and  thus,  Pt/p0=(Rt/R0)(St/S0)(l0/lt) 


where  7  is  the  coefficient  of  the  linear  expansion  of  copper 
for  rise  of  temperature.     Hence 


28          ELECTRIC  CABLES  AND  NETWORKS 


Pt  —Po  {  l  +(a+7)*  }  approximately. 
We  also  have  P\=Rt(m/L\)  and  p'0=R0(m/L*0), 
and  thus,  p't=P'0(Rt/R0)/(L\/L\) 


=p'0  {  1  +  (  a  —  2y)t  }  approximately. 
If  a=  0-00420  and  y=  0-000017,  we  may  write 

^=^(1+0-004220  and  p't=  p'0(l+W041M). 

It  is  customary  to  assume  that  the  values  of  the  temper- 
ature coefficients  for  Rt,  pt  and  p't  are  the  same.  We  see 
that  the  maximum  error  which  arises  from  this  neglect  of 
the  thermal  expansion  of  the  metal,  when  t  is  less  than 
50°  C.,  is  less  than  0-2  per  cent. 
Numerical  ^-s  an  application  of  the  formulae  we  shall 

example    consicier  the  problem  of  finding  the   percentage 
conductivity  in  terms  of  "  standard  "  copper  of  a  copper 
rod  one  centimetre  in  diameter.     We  shall  suppose  that 
when  a  current  is  flowing  through  it,  the  potential  difference 
between  two  knife  edges  at  a  distance  of  200-0  cms.  apart 
as  read  by  an  accurately  calibrated  millivoltmeter  is  0-1948 
volt.     We  shall  also  suppose  that  the  temperature  of  the 
rod  is  35-6°  C.  and  that  the  current  flowing  through  it  and 
the  voltmeter  in  parallel,  as  read  by  a  Kelvin  balance  is 
398-9    amperes.     If    the    resistance    of    the    millivoltmeter 
with  its  connecting  leads  be  7-30  ohms  the  current  flowing 
through  it  will  be  0-1948/7-3,  or  0-027  ampere  nearly.     The 
current  in  the  conductor  may  therefore  be  taken  as  398-9 
amperes,  and   hence,  the  resistance  between  the  two  equi- 
potential  surfaces  passing  through  the  points  of  contact  of 
the  knife  edges  will  be  0-1948/398-9,  that  is,  0-0004883  ohm. 
If  we    assume  that   the  lines   of  flow   of  the   current  are 
parallel  to  the  axis  of  the  conductor  so  that  the  equipotential 
surfaces  are  planes  perpendicular   to   this  axis,  then  since 
0-7854  is  the  area  of  the  cross  section  of  the  rod,  we  have 


CONDUCTIVITY  29 

=0-0004883  x  0-7854/200 

=  1-917  microhms, 
and   therefore, 

^15.6  =  1.917(1  +0-0042  x  15-6)/(l  +0-0042x35-6) 

=  1-917/1-079 

=  1-777. 

Now  at  this  temperature  the  standard  volume  resistivity 
is  1-731.  The  percentage  conductivity  of  the  copper 
forming  the  rod  is,  therefore,  100x1-731/1-777,  that  is, 
97-4.  As  a  2  per  cent,  variation  from  the  adopted 
standard  is  considered  permissible  by  manufacturers  and 
engineers,  this  conductor  would  legally  satisfy  a  specifica- 
tion insisting  on  a  99  per  cent,  conductivity  but  not  one 
insisting  on  a  100  per  cent,  conductivity.  In  practice  many 
tests  must  be  taken,  and  the  conditions  of  the  experiment  or 
the  method  adopted  must  be  varied  in  some  of  the  tests,  be- 
fore the  experimenter  can  make  certain  that  his  maximum 
inaccuracy  is  less  than  the  half  of  1  per  cent.  It  has  to  be  re- 
membered that,  since  the  resistance  is  found  by  dividing  the 
reading  of  the  millivoltmeter  by  the  reading  of  the  ammeter, 
the  percentage  error  in  the  computed  resistance  is  sometimes 
equal  to  the  sum  of  the  two  instrumental  percentage  errors. 
Some  of  the  substances,  sulphur  for  instance, 

Tinning 

in  the  materials  used  to  insulate  copper  wires, 
attack  copper.  When,  therefore,  these  substances  are  used 
the  wires  are  given  a  coating  of  pure  tin.  As  the  conduct- 
ivity of  tin  is  less  than  that  of  copper,  the  conductivity  of 
a  tinned  copper  conductor  will  be  slightly  less  than  that  of 
a  pure  copper  conductor  of  the  same  diameter.  For  this 
reason  the  conductivity  of  all  tinned  copper  conductors 
whose  diameters  lie  between  0-104  and  0-028  inches  (No. 
12  and  No.  22  S.W.G.)  inclusive,  is  allowed  to  be  1  per 
cent,  lower  than  that  of  pure  copper. 


30 


ELECTRIC   CABLES   AND  NETWORKS 


Lines  of 
flow 


In    measuring    a    resistance    by  the  fall  of 
potential    method,    as    described    in    the    last 

section,  we  assumed 
that  the  lines  of  flow 
of  the  current  were 
straight  lines.  If  the 
metal  is  not  homo- 
geneous or  if  its 
diameter  vary  ap- 
preciably, this  as- 
sumption is  not  per- 
missible and  errors 
may  arise  from  this 
cause.  For  example, 
in  measuring  the  re- 
sistance of  the  cop- 
per bonds  used  for 
rails  in  electric  trac- 
tion the  result  de- 
pends on  the  equi- 
potential  surfaces 
chosen.  This  is 
illustrated  in  Fig.  1. 
Here  ab  is  a  short 
cylindrical  copper 
conductor  connect- 
ing two  large  copper 
cylinders  A  and 
B.  The  lines  and 
arrow-heads  indicate 
the  direction  of  the 
flow  of  the  current 
through  the  conduc- 


CONDUCTIVITY  31 

tors.  The  curved  lines  AA±,  BB^  aa^  and  bb  1?  cutting  the 
lines  of  flow  at  right  angles  are  sections  of  equipotential 
surfaces  by  the  plane  of  the  paper.  If  we  were  to  put 
the  knife  contacts  at  A  and  B,  and  proceed  as  in  the 
last  section  but  one,  the  resistance  measured  would  be 
that  between  the  surfaces  AA±,  and  BB^  and  unless  these 
surfaces  were  accurately  known  and  also  the  temperatures 
at  the  various  sections,  it  would  be  impossible  to  deduce 
the  conductivity  of  the  copper.  Similarly  if  the  knife 
edges  were  placed  at  a  and  b  we  would  get  another  value  of 
E,  but  as  the  equipotential  surfaces  are  still  appreciably 
curved  an  error  would  be  introduced  if  we  made  calculations 
on  the  assumption  that  they  were  planes. 
Measuring  ^n  ^e  rePor^  °^  the  Standardization  Com- 
o^tempera-  m^^ee  °f  the  American  Institution  of  Electrical 
ture  Engineers  (Journal,  1907),  it  is  recommended 
that  the  rise  of  temperature  in  all  conductors  should,  when 
practicable,  be  determined  by  their  increase  of  resistance. 
The  resistance  may  be  measured  either  by  a  Wheat- 
stone's  bridge  method  or  preferably  by  an  ammeter  and 
voltmeter.  The  temperature  calculated  in  this  way  is 
usually  higher  than  that  obtained  by  placing  a  thermometer 
against  the  conductor.  It  is  also  recommended  that,  when 
a  thermometer  is  placed  against  the  surface  of  the  object 
of  which  the  temperature  is  being  measured,  the  bulb  should 
be  covered  by  a  pad  of  definite  area.  For  instance,  a  con- 
venient pad  may  be  made  of  cotton  waste  contained  in  a 
shallow  circular  box  about  1J  in.  in  diameter.  The  bulb 
of  the  thermometer  is  inserted  through  a  hole  in  the  side 
of  the  box.  If  the  pad  be  too  large  it  interferes  with  the 
natural  radiation  of  heat  from  the  metal  surface  and  thus 
introduces  complications  into  the  test. 


32          ELECTRIC  CABLES  AND  NETWORKS 

The  formula 

B=B0(l+OW42t)  ........     (1), 

which  shows  the  connexion  between  the  resistance  E  of 
copper  at  t°  C.,  and  its  resistance  R0  at  0°  C.,  enables  us  to 
calculate  the  temperature  rise  when  the  values  of  the 
initial  and  final  resistances,  R  and  R',  are  known.  If 
R'  be  the  resistance  when  the  temperature  is  t  +  x,  we 
have  by  (1) 


Hence  from  (1)  and  (2), 

E'/E  =  {  1  +0*0042(£  +x)  }  /  {  1  +0-0042*  } 
=l+42»/(lp,000-t-42*), 

and  therefore,  x=  (238+t)(R'/R—l)  ........     (3). 

As  an  example  of  the  use  of  (3),  let  us  suppose  that  R  is 
81-8  ohms  at  the  initial  temperature  of  12°  C.,  and  that  the 
resistance  is  finally  85-8  ohms.  By  (3),  we  have 

a=(238  +  12)(86-8/81-8—  1) 
=1,000/81-8 
=  12-2°  C. 

As  another  example  let  us  take  the  case  of  an  armature 
winding.  Before  the  test  let  us  suppose  that  its  resistance 
was  0*230  of  an  ohm  and  that  the  temperature  was  25°  C., 
and  that  after  carrying  a  current  for  some  time  the  re- 
sistance rises  to  0*271  of  an  ohm.  In  this  case 

x—  (238+25)(0*271/0*23—  1) 
=  46-9°  C. 

J.  Dewar  and  J.  A.  Fleming  (Phil.  Mag.,  [5], 
coeffideiS6    vol.  36>  P-  27  1,  1893)  give  the  following  values 
of  the  mean    temperature    coefficients  of  pure 
metals  for  temperatures  from  0°  to  100°  C. 


CONDUCTIVITY 


33 


Metal 

a 

Metal 

a 

Aluminium    . 
Cadmium 
Copper     .... 
Gold 

0-00435 
0-00419 
0-00428 
0-00377 

Nickel    .... 
Palladium    . 
Platinum     .      .      , 
Silver     . 

0-00622 
0-00354 
0-00367 
0-00400 

Iron    
Lead         .... 

0-00625 
0-00411 

Thallium      . 
Tin   .      .      .      .      . 

0-00398 
0-00440 

Masrnesium 

0-00381 

Zinc 

0-00406 

Stranded 
cables 


It  is  interesting  to  notice  that  the  temperature  coefficient 
of  platinum  is  practically  the  same  as  the  temperature  co- 
efficient of  the  pressure  of  a  gas  at  constant  volume. 

When  the  area  of  the  cross  section  of  the 
copper  in  a  cable  has  to  be  greater  than  6,400 
circular  mils,  that  is,  than  the  area  of  the  cross  section  of 
a  solid  cylindrical  conductor  0-08  of  an  inch  in  diameter 
(No.  14  S.W.G.)  it  is  customary  to 
form  the  conductor  of  several  strands 
of  wire.  In  general  there  is  one  cen- 
tral wire  and  round  this  wire  is  a  layer 
of  six  wires,  and  after  this  the  number 
of  wires  in  successive  layers  increases 
in  arithmetical  progression,  the  com- 
mon difference  being  6.  The  number 
of  strands,  for  instance,  in  the  section 
of  the  cable  shown  in  Fig.  2  is  1+6+12,  that  is,  19. 

When  there  are  n  layers,  the  total  number  N  of  strands 
is  given  by 


Thus 


FIG.   2.— Nineteen 
strand  cable. 


=(n+l)2—  n—  2/3 


34          ELECTRIC   CABLES   AND   NETWORKS 


FIG.  3. — Cross  section  of  a 
cable  containing  37 
strands  of  wire.  The 
middle  wire  is  straight, 
and  consecutive  layers 
are  spiralled  in  oppo- 
site directions. 


Hence  (N/3)1'2  is  greater  than  n  but  less  than  n+1.     Con- 
sequently the  number  of  layers  in  a  cable  of  N  strands  is 

the  integral  part  of  (JV/3)1/2.  For 
example  if  N  were  331,  the  num- 
ber of  layers  would  be  10,  for 
(N/3)V*  equals  (110-3.  .)1/2,  and 
the  integral  part  of  this  radi- 
cal is  obviously  10.  We  should 
therefore  have  ten  layers  contain- 
ing 6,  12,  18  . .  60  wires  in 
addition  to  the  central  wire. 

In  ordinary  cables  the  number 
of  strands  used  are  1,  7, 19,  37,  61, 
91  or  127.  A  cable  consisting  of 
N  strands  of  No.  M  wire  is  called 

an  N/M  cable. 

In  Fig.  3  the  cross  section  of  a  cable  consisting  of  37  strands 

is  shown.     It  will  be  noticed  that  after  the  first  layer  the 

sections  of   the   strands  do   not 

necessarily    touch    the    sections 

adjacent  to  them.     In  practice, 

consecutive  layers  of  the  strands 

are  given  a  slight  twist  in  oppo- 
site directions,   the  effect  being 

that  the  centres  of  the  sections 

of  the  strands  in  each  layer  lie 

on  a  circle  concentric  with  the 

section     of     the     central    wire. 

Since    the    wires   in   the   layers 

are  helical,  their   sections   by  a 

plane  perpendicular  to  the  axis 

of  the  cable  will  not  be  exactly  circular. 

If  all  the  strands  were  parallel  circular  cylinders  and  if 


FIG.  4. — Cross  section  of  a 
stranded  cable  of  37 
wires  when  the  wires  are 
all  parallel.  Notice  that 
the  difference  between 
the  numbers  of  wires  in 
consecutive  layers  is  six. 


CONDUCTIVITY 


35 


the  cable  had  to  be  as  compact  as  possible  the  section  would 
be  hexagonal  in  shape  (Fig.  4),  and  every  conductor  inside 
the  outside  layer  would  touch  the  six  adjacent  conductors. 


FIG.  5. — Stranded  cable.     The  strands  in  successive  layers  are  spiralled 
in  opposite  directions. 

The  effect  of  giving  a  helical  form  to  the  layers  is  to  make 
them  bind  together.  The  inner  and  outer  boundaries  of  the 
layers  (Fig.  5)  touch  concentric  cylindrical  surfaces.  The 
radius  of  the  inner  cylindrical  surface  which  every  wire  on 
the  nth  layer  touches  is  (2n — l)r. 

We  shall  now  consider  the  number  of  strands  it  would  be 
possible  to  get  on  the  nth  layer  on  the  assumption  that 
the  sections  of  the  strands  are  circles.  In  Fig.  6,  let  r  be 


FIG.  6. 


the  radius  of  each  of  the  small  circles  which  touch  one 
another  and  the  large  circle.  Let  the  radius  of  the  large 
circle  be  (2n — l)r.  The  angle  c/>  subtended  at  the  centre  of 


36 


ELECTRIC   CABLES   AND   NETWORKS 


the  large  circle  by  the  line  joining  the  centres  of  the  two 
small  ones  is  given  by 

sin(<£/2)  =  r/2nr  —  l/2n. 

Thus  the  number  of  the  wires  that  would  go  round  the  large 
circle  is  the  integral  part  of  the  function, 

ISO/sin"^  l/2n), 

the  angle  sirr~1(l/2n)  being  measured  in  degrees.  The 
values  of  this  function  for  various  values  of  n  are  given  in 
the  following  table. 


n 

1 

2 

3 

4 

5 

6 

7 

Function  . 

6 

12-4 

18-8 

25-1 

31-4 

37-7 

43-98 

The  numbers  are  approximately  in  arithmetical  progression, 
the  common  difference  being  very  nearly  2?r  or  6-28  .  .  ,  when 
n  is  greater  than  2.  Thus  theoretically  we  could  put  6,  12, 
18,  25,  31  .  .  wires  in  the  successive  layers  instead  of 


the  6,  12,  18,  24,  30 


FIG.  7. — Section  of  a  27 
strand  cable  having 
a  central  core  of 
three  strands. 


used  in  practice. 

Cables  are  occasionally  made  hav- 
ing a  central  core  formed  of  three 
strands  (Fig.  7),  or  more  rarely  of 
four  strands.  In  the  case  when  the 
core  has  three  strands,  the  number 
of  strands  in  the  ftth  layer  is  6^  +  3, 
and  the  total  number  N  of  strands 
is  given  by 

,    .    .    +6^  +  3 


Thus  the  number  of  layers  is  one  less  than  \/ZV/3. 

Proceeding  as  before,  we  find  that  the  number  of  wires  in 
the  nth  layer  is  the  integral  part  of  the  function 


CONDUCTIVITY 
180 


37 


where  the  angle  is  expressed  in  degrees. 

The  values  of  this  function  are  given  in  the  following 
table.     It  will  be  seen  that  when  n  is  greater  than  3  the 


n 

1 

2                  3 

4                  5 

6 

Function  . 

9-7 

16-1 

22-5 

28-8         35-1 

41-4 

numbers  practically  form  an  arithmetical  progression,  the 
common  difference  being  approximately  2-n-  or  6-28.  .  . 
Theoretically,  therefore,  it  is  possible  to  put  16  instead  of  15 
wires  in  the  second  layer  and  35  instead  of  33  on  the  fifth. 

Effect  of  When  a  stranded  cable  has  a  central  wire  the 

lay  on  the 
mass  of       axis  of  this  wire  is  a  straight  line  but  the  axis 

the 
conductor     of  every  other  wire  of  the  cable  forms  a  helix, 

all  the  helices  forming  a  layer  having  practically  the  same 

pitch.     If  a  point  moving  along  the  helical 

axis  of  a  strand  of   the    cable    make    a 

complete   revolution    round    the   central 

wire  when  it  has   advanced  a   distance 

parallel  to  this  wire  equal  to  n  times  the 

diameter  of  the  helix,  the  wire  is  said  to 

have  a  lay  of  1  in  n.     If  a  be  the  angle 

which  the  tangent  LP  at  any  point  L  of 

the    helix    (Fig.   8)    makes   with    a   line 

through  L  parallel  to  the  central  wire,  if 

LN  =  nd=l,  where  d  is  the  diameter  of 

the  helix,  and  if  NP  be  at  right  angles 

to  LN,  then  LP  will  be  the  length  x  of  the 

helical  wire  corresponding  to  a  length  I  of  the  central  wire, 

and    PN  will  equal   -nd.      Hence    tana  =  ird/l  =  ir/n,  and 

x  =  lsec<t  =  l(l+7T*/n2)l/2.     Since   I  is    the   pitch    of  the 


FlG   8 


38 


ELECTRIC  CABLES  AND  NETWORKS 


helix  and  equals  nd,  we  see  that  the  pitch  of  the  helical 
strands  in  the  various  layers  varies  as  the  diameters  of  these 
layers,  provided  that  the  lay  is  the  same  for  all  the  strands. 
We  shall  now  consider  the  effect  of  the  lay  on  the  mass  of 
the  copper  required.  Let  us  take  the  case  of  a  7-strand  cable 
of  length  I  and  let  us  find  the  factor  by  which  the  mass  of  the 
central  wire  has  to  be  multiplied  in  order  to  get  the  mass  of 
the  whole  copper  in  the  length  /.  By  the  formula  given 
above  the  length  of  the  six  helical  wires  in  the  first 
layer  is  Z(l-|-7r2/ft2)1/2.  Hence  the  required  multiplier  is 


For  example,  if  the  lay  were  1  in  20,  n  would  be  20  and 
the  multiplier  would  be  7*0736.  If  the  wires  were  straight 
the  multiplier  would  be  7,  and  thus,  the  effect  of  the  lay 
is  to  increase  the  mass  of  the  conductor  by  a  little  more 
than  1  per  cent. 

Some  manufacturers  use  a  lay  as  low  as  1  in  12  and  others 
as  high  as  1  in  30.  The  value  usually  taken  is  1  in  20. 

In  the  following  table  the  factors  for  multiplying  the  mass  of 
a  strand  equal  in  length  to  the  cable  in  order  to  get  the  mass  of 
the  conductor  are  given  for  lays  of  1  in  12,  1  in  20  and  1  in  30. 


No.  of 
Strands 

Multiplier 

Lay  of  1  in  12 

Lay  of  1  in  20 

Lay  of  1  in  30 

3        .       .      .... 

3-101 

3-037 

3-016 

4       ... 

4-135 

4-049 

4-022 

7       .      .   •  . 

7-202 

7-074 

7-033 

12       ... 

12-404 

12-147 

12-066 

19       ... 

19-607 

19-221 

19-098 

37       ... 

38-213 

37-661 

37-198 

61        ... 

63-022 

61-736 

61-328 

91       .      ,      . 

94-033 

92-103 

91-492 

The  cable  with  twelve  strands  has  a  core  of  three  strands. 


CONDUCTIVITY 


39 


Effect  of  lay        Let  us  now  consider  how  the  resistance  of  a 

resistance      stranded  cable  varies  with  the  lay  of  the  wires. 

conductor  As  the  greater  the  lay  of  the  wires,  the  greater 
the  mass  of  the  conductor  in  a  given  length  of  the  cable,  it 
might  at  first  sight  be  thought  that  the  resistance  would 
diminish  as  the  lay  is  increased.  If  we  remember,  how- 
ever, that  the  great  bulk  of  the  lines  of  flow  of  the  current  in 
the  strands  must  follow  helical  paths  we  should  expect  that 
the  resistance  of  all  these  paths  in  parallel  will  be  greater 
than  the  resistance  of  the  shorter  paths  when  the  strands 
are  straight  and  this  is  found  to  be  the  case  in  practice. 

In  a  7-strand  cable,  for  example,  if  r  be  the  resistance 
of  the  central  strand,  the  resistance  of  the  other  six  strands 
in  parallel  will  be  r(l-f-77-2/7i2)1/2/6,  where  n  is  the  lay  of 
the  cable,  assuming  that  there  is  no  flow  of  current  from  one 
strand  to  another.  Hence  the  factor  by  which  the  resistance  of 
the  central  wire  has  to  be  multiplied  by  in  order  to  get  the  re- 
sistance of  the  cable  is  I/  { 1  +6/(l  +  Tr2/^2)1/2 } ,  and  this  is  al- 
ways greater  than  one-seventh.  Hence  the  effect  of  the  twist- 
ing is  to  increase  the  resistance  of  the  cable  per  unit-length. 

In  the  following  table  the  factors  for  multiplying  the 
resistance  of  a  single  strand  equal  in  length  to  the  cable  in 
order  to  get  the  resistance  of  the  cable  are  given  for  lays  of 
1  in  12,  1  in  20,  and  1  in  30. 


No.  of 
Strands 

Multiplier 

Lay  of  1  in  12 

Lay  of  1  in  20 

Lay  of  1  in  30 

3  .... 

0-34457 

0-33742 

0-33516 

4  .... 

0-25843 

0-25307 

0-25137 

7  .... 

0-14696 

0-14436 

0-14353 

12  .... 

0-08614                   0-08436 

0-08379 

19  .      .       .      . 

0-05431                    0-05324 

0-05290. 

37  .... 

0-02791                   0-02735 

0-02717 

61  .... 

0-01694                   0-01659 

0-01648 

91  .      ,      .      . 

0-01136 

0-01112 

0-01105 

40 


ELECTRIC  CABLES  AND  NETWORKS 


Permissible 

current  in 

cables 


In  the  wiring  rules  (1907)  of  the  English 
Institution  of  Electrical  Engineers  the  following 
table  of  the  maximum  permissible  currents  for 
copper  conductors  laid  in  casing  or  tubing  is  given.  The 
maximum  currents  may  be  calculated  from  the  formula 


where  C  is  the  current  in  amperes  and  S  is  the  sectional  area 
in  thousandths  of  a  square  inch. 


Number  and 
Gauge  of  Strands. 
S.W.G.  or  Ins. 

Section 
Sq.  Ins. 

Mass 
Lbs.  per 
1,000  yds. 

Max. 
Current 
Amps. 

Current 
Density 
Amps,  per 
sq.  in. 

Length 
in  yards 
for  1 
volt  drop 

3/25 

0-00092 

11-12 

2-5 

2,800 

15 

3/24 

0-00112 

13-45 

2-9 

2,600 

16 

3/23 

0-00133 

16-01 

3-3 

2,500 

17 

1/18 

0-00181 

20-92 

4-2 

2,300 

18 

3/22 

0-00181 

21-79 

4-2 

2,300 

18 

7/25 

0-00216 

25-87 

4-9 

2,250 

18 

3/21 

0-00237 

28-45 

5-3 

2,250 

19 

1/17 

0-00246 

28-48 

5-4 

2,200 

19 

7/24 

0-00262 

31-29 

5-7 

2,200 

19 

3/20 

0-00299 

36-02 

6-4 

2,150 

19 

7/23 

0-00311 

37-24 

6-6 

2,150 

20 

1/16 

0-00322 

37-20 

6-8 

2,100 

20 

3/19 

0-00370 

44-47 

7-6 

2,050 

20 

1/15 

0-00407 

47-08 

8-2 

2,000 

21 

7/22 

0-00424 

50-70 

8-5 

2,000 

21 

1/14 

0-00503 

58-12 

9-8 

1,950 

21 

3/18 

0-00532 

64-02 

10-3 

1,950 

21 

7/21 

0-00554 

66-21 

11-0 

1,950 

21 

7/20 

0-00701 

83-81 

13-0 

,850 

22 

7/19 

0-00865 

103-5 

15-0 

,750 

24 

7/18 

0-01250 

149-0 

21-0 

,700 

25 

7/17 

0-017 

202-8 

27-0 

,600 

26 

19/20 

0-019 

228-0 

29-0 

,550 

27 

7/16 

0-022 

264-8 

33-0 

1,500 

28 

19/19 

0-023 

281-0 

35-0 

1,450 

28 

CONDUCTIVITY 


41 


Number  and 
Gauge  of  Strands. 
S.W.G.  or  Ins. 

Section 
Sq.  Ins. 

Mass 
Lbs.  per 
1,000  yds. 

Max. 
Current 
Amps. 

Current 
Density 
Amps, 
per  sq.  in. 

Length 
in  yards 
for  1 
volt  drop 

7/0-068  in. 

0-025 

299-0 

36-0 

1,450 

29 

7/15 

0-028 

335-0 

40-0 

1,450 

29 

19/18 

0-034 

405-0 

47-0 

1,400 

30 

7/14 

0-035 

414-0 

48-0 

1,400 

30 

19/17 

0-046 

551-0 

60-0 

1,300 

32 

7/0-095  in. 

0-050 

584-0 

65-0 

1,300 

32 

19/0-058  in. 

0-050 

591-0 

65-0 

1,300 

32 

19/16 

0-060 

720-0 

75-0 

1,250 

33 

19/15 

0-075 

911-0 

91-0 

1,200 

35 

19/14 

0-094 

1125 

108-0 

1,150 

36 

19/0-082  in. 

0-100 

1182 

113-0 

1,150 

36 

37/16 

0-117 

1403 

130-0 

1,100 

37 

19/13 

0-125 

1488 

136-0 

1,100 

38 

37/15 

0-150 

1776 

157-0 

1,100 

39 

19/0-101  in. 

0-150 

1793 

155-0 

1,050 

40 

37/14 

0-180 

2192 

187-0 

1,050 

40 

37/0-082  in. 

0-20 

2303 

200-0 

1,000 

40 

37/0-092  in. 

0-25 

2900 

238-0 

950 

42 

37/0-101  in. 

0-30 

3494 

280-0 

950 

43 

37/0-  110  in. 

0-35 

4145 

320-0 

900 

45 

61/13 

0-40 

4781 

350-0 

900 

47 

61/0-098  in. 

0-45 

5425 

380-0 

850 

47 

61/0-101  in. 

0-50 

5762 

425-0 

850 

47 

61/0-  108  in. 

0-55 

6588            450-0 

800 

48 

61/0-llOin. 

0-60 

6836            490-0 

800 

48 

61/0-118in. 

0-65 

7865 

530-0 

800 

50 

91/0-098  in. 

0-70 

8094 

550-0 

800 

50 

91/0-101  in. 

0-75 

8597 

590-0 

800 

50 

91/12 

0-80 

9115             625-0 

800 

51 

91/0-  110  in. 

0-90 

10200 

670-0 

750 

53 

91/0-118in.         1-00 

11730             750-0 

750 

54 

127/0-101  in.         I'OO 

12000             750-0 

750 

55 

It  is  to  be  noticed  that  the  symbol  19/0-058  in.  stands 
for  a  conductor  of   19  strands  of   wire,  the    diameter   of 


42 


ELECTRIC  CABLES  AND  NETWORKS 


each  of  which  is  0-058  inches.  The  sizes  3/25,  3/24, 
3/23  are  the  usual  sizes  of  the  conductors  used  in 
electric  light  fittings.  It  is  worth  remembering  that 
when  the  current  density  is  1,000  amperes  per  square 
inch  the  pressure  drop  is  1  volt  for  40  yards.  At  this 
current  density,  for  instance,  if  the  going  and  return  con- 
ductors are  each  40  yards  long  the  difference  of  pressure 
between  the  far  ends  of  the  conductors  will  be  2  volts  less 
than  that  between  the  ends  where  they  are  joined  to  the 
switchboard.  The  difference  in  the  two  values  of  the 
pressure  is  the  pressure  required  owing  to  the  resistance  of 
the  conductors.  In  practice  the  maximum  permissible 
value  of  the  current  in  conductors  is  fixed  by  the  voltage 
drop,  and  not  by  the  rise  of  temperature  of  the  conductor. 
By  the  Board  of  Trade  Rules,  the  pressure  at  any  con- 
sumer's terminals  must  not  vary  by  more  than  4  per  cent, 
from  the  declared  constant  pressure,  and  this  regulation 
generally  necessitates  a  low  current  density  in  the  mains. 

The  In   the   following   table,   compiled  from   the 

of  cables  wiring  rules  (1907)  of  the  Institution  of  Elec- 
trical Engineers  the  resistances  at  60°  F.  of  copper  con- 
ductors per  1,000  yards  are  given  in  ohms. 


Gauge 
S.W.G.  or  ins. 

Res. 

Gauge 
S.W.G.  or  ins. 

Res. 

Gauge 
S.W.G.  or  ins. 

Res. 

1        I 

3/25 

26-01     7/23     7-721     7/18      1-930 

3/24 

21-50     1/16     7-473    7/17      1-418 

3/23 

18-07      3/19     6-504    19/20 

1-267 

1/18 

13-29      1/15    ,  5-905     7/16 

1-086 

3/22 

13-27      7/22 

5-672    19/19 

1-026 

7/25 

11-12  .|    1/14 

4-783    7/0-068  in. 

0-962 

3/21 

10-16      3/18 

4-516 

7/15 

0-858 

1/17 

9-761     7/21     4-343    19/18 

0-713 

7/24 

9-190     7/20     3-431     7/14 

0-695 

3/20 

8-029    7/19 

2-779 

:  19/17 

0-523 

i 

CONDUCTIVITY 


43 


Gauge 
S.W.G.  or  ins. 

Res. 

Gauge 

S.W.G.  or  ins. 

Res. 

Gauge 
S.W.G.  or  ins. 

Res. 

7/O095in. 

0-493 

19/0-101 

0-161 

61/0-108  in. 

0-044 

19/0-058  in. 

0-488 

37/14  • 

0-132 

61/0-110  in. 

0-042 

19/16 

0-401 

37/0-082  in.  0-125 

61/0-118  in. 

0-037 

19/15 

0-317 

37/0-092  in. 

0-100 

91/0-098  in. 

0-036 

19/14 

0-257 

37/0-101  in. 

0-083 

91/0-101  in. 

0-034 

19/0-082  in. 

0-244 

37/0-llOin. 

0-070 

91/12 

0-032 

37/16 

0-206 

61/13       0-061 

91/0-110  in. 

0-028 

19/13 

0-194 

61/0-098  in.  0-053 

91/0-118  in. 

0-025 

37/15 

0-163 

61/0-101  in.  0-050 

127/0-101  in. 

0-024 

High  fre-  With  alternating  currents  having  a  frequency 
alternating  not  greater  than  50  an(l  w^n  conductors  the 

currents  diameters  of  which  are  not  greater  than  a  centi- 
metre, the  above  formulae  can  be  used  for  calculating  the 
resistance.  When,  however,  the  frequency  of  the  alterna- 
tions is  high  or  the  diameter  of  the  conductor  is  large,  the 
effective  resistance  to  the  alternating  currents  is  greater  than 
for  direct  currents.  The  reason  for  this  is  that  the  current 
starts  at  the  surface  of  the  wire  and  takes  time  to  penetrate 
into  the  interior. 

Let  us  consider,  for  example,  the  currents  in  a  concen- 
tric main,  that  is,  a  main  formed  by  a  solid  or  a  hollow 
copper  cylinder  surrounded  by  a  cylindrical  tube.  With 
high  frequencies  the  going  and  return  currents  distribute 
themselves  in  such  a  way  that  practically  no  magnetic 
forces  are  produced  in  the  solid  copper.  There  is  a  con- 
centration of  the  current  on  the  outside  of  the  inner  con- 
ductor and  on  the  inside  of  the  outer  conductor  and  this 
considerably  increases  the  effective  resistance  of  the  con- 
ductors. Let  R  and  Ea  denote  the  resistances  of  the  inner 
conductor  of  a  concentric  main  to  direct  and  to  alternating 
currents  respectively.  If  W  is  the  power  expended  on  the 
inner  conductor  when  a  current  A  of  the  given  frequency  is 


44          ELECTRIC  CABLES   AND  NETWORKS 

flowing  in  it,  Ra  equals  W/A2.  The  value  of  Ra/R  can  be 
found  from  the  following  table  which  is  practically  the  same 
as  that  first  given  by  Lord  Kelvin.  In  this  table  /  denotes 
the  frequency,  in  cycles  per  second,  a  the  radius  of  the  con- 
ductor in  centimetres,  and  p  its  volume  resistivity,  in  C.G.S. 
measure,  so  that  p  is  1,000  times  the  value  of  the  volume 
resistivity  in  microhms. 


Value  of  27TO, 

0-0  ..........  1-000 

0-5  ..........  1-000 

1-0  ..........  1-005 

1-5  ..........  1-026 

2-0  ..........  1-078 

2-5  ..........  1-175 

3-0  ..........  1-318 

3-5  ..........  1-492 

4-0  ..........  1-678 

4-5  ..........  1-863 

5-0  ..........  2-043 

5-5  ..........  2-219 

6-0  ..........  2-393 

7-0  ..........  2-743 

8-0  ..........  3-096 

9-0  .  .........  3-447 

10-0  .  .........  3-798 

15-0  ..........  5-562 

20-0  ..........  7-328 

30-0  ..........  10-86 

n-0  .  .          .  .          .  .          .  .          .  .  0-3535  n  approx. 

for  values  of  n  greater  than  30. 

When  n  is  greater  than  30,  Ra/R  =  n/2  vx2  :=  n  x  0-3535 
approximately. 

Hence  Ra=(n/2v2) 


We  see  therefore  that  the  value  of  Ra  is  the  same  as  if 
the  current  were  uniformly  distributed  over  a  thin  skin  on 
the  surface  of  the  conductor  of  thickness  (l/27r)  ^p/f.  We 
see  also  that  with  very  high  frequencies  the  thicker  the 


CONDUCTIVITY  45 

wire  the  less  is  the  resistance  per  unit  length  it  offers  to 
alternating  currents,  but  whilst  with  direct  currents  the 
resistance  varies  inversely  as  the  square  of  the  diameter  of 
the  wire,  with  high  frequency  alternating  currents  it  varies 
inversely  as  the  diameter. 

Data  for  ^-n   calculations   in   connexion     with    copper 

calculations     cabieS)  the  following  data  will  be  found  useful. 
Standard  Annealed  High  Conductivity  Copper  at  60°  F. 
Volume  resistivity  (cubic  cm.)  =1-696  microhms. 

(cubic  inch)  =0-6679 
Resistance  per  mile  =0-04232/$  ohms,  where  S  is  the 

area  of  the  cross  section  in  square  inches. 
Resistance  per  yard      =0-00002404/$  ohms. 
Mass  resistivity  =0-1508. 

Resistance  per  mil  foot  =  10-20  ohms. 
Mass  per  mile  in  Ibs.      =20350  8. 
Mass  per  yard  in  Ibs .      =11-56$. 

Standard  Hard-drawn  High  Conductivity  Copper  at  60° F. 
Volume  resistivity  (cubic  cm.)=  1-731. 

(cubic  inch)  =  0-6813. 
Resistance   per  mile=0-04317/$,  where  S  is  the  area 

of  the  cross  section  in  square  inches. 
Resistance  per  yard       =0-000024537$  ohms. 
Mass  Resistivity  =0-1539. 

Resistance  per  mil  foot  =10-41  ohms. 
Mass  per  mile  in  Ibs.       =20350  8. 
Mass  per  yard  in  Ibs.      =11-56$. 

REFERENCES. 

G.  F.  C.  Searle,  Experimental  Elasticity. 

J.  A.  Fleming,  A  Handbook  for  the  Electrical  Laboratory  and  Testing 

Room,  vol.  i. 
A.  Matthiessen  &  M.  von  Bose,  "  On  the  Influence  of  Temperature 

on  the   Electro   Conducting  Power  of  Metals."     Phil.   Trans. 

p.  1,  vol.  152,  1862. 


46          ELECTRIC  CABLES   AND  NETWORKS 

J.  Dewar  &  J.  A.  Fleming,  "  The  Electrical  Resistance  of  Metals  and 

Alloys  at  Temperatures  approaching  the  Absolute  Zero."     Phil. 

Mag.  p.  299,  Sept.  1893. 
J.  W.  Swan  &  J.  Rhodin,  "  Measurements  of  the  Absolute  Resistance 

of  Pure  Electrolytic  Copper."     Proc.  Roy.  Soc.  p.  64,  1894. 
T.  C.  Fitzpatrick,  '•  On  the  Specific  Resistance  of  Copper   and   of 

Silver,"  p.   131,  B.  A.  Report,  1894. 
"  Report  of  the  Committee  on  Copper  Conductors."     Journal  of  the 

Inst.  of  Elect.  Eng.  vol.  xxix,  p.  169,  1900. 
"  British  Standard  Tables  of  Copper  Conductors."     Reports  issued 

by  the  Engineering  Standards  Committee. 
Lord  Kelvin,  Journ.  of  the  Inst.  of  Elect.  Engin.,  vol.  xviii,  p.  4. 

"  Ether,  Electricity  and  Ponderable  Matter."   Jan.  1889.     Also, 

Mathematical  and  Physical  Papers,  vol.  3,  p.  493. 
G.  F.  C.  Searle,  "  On  the  Elasticity  of  Wires."     Phil.  Mag.  [5],  vol. 

xlix,  p.  193,  1900,  or  p.  112,  Experimental  Elasticity. 


INSULATIVITY 


CHAPTER  in 

Insularivity 


IJT  order  to  present  leakage  of 

an.  CKCVZ1C  CQllflOCtflr  IK  BB  H6CC8BaTV  to 

in  a  snitaUe  iaMJuiiMMiiiiiil     The 


For  low 


to  a  ftnrof 

!••!•  •  ifaiatiiM 
it,  that  is, the  efectne  JJM.W*^  is  the 


"  T    ^.i_     l-^lr     T^ 


50 


ELECTRIC   CABLES   AND  NETWORKS 


opposite  faces.  It  is  thus  the  same  as  the  volume  resistivity 
p.  It  is,  however,  convenient  to  use  a  different  symbol  as  it 
is  customary  to  measure  a  in  megohms  and  p  in  microhms, 
and  hence, 

p=a.  1012. 

Sir  William  Preece  defines  the  specific  insulation  a'  of  a 
dielectric  as  the  resistance  in  megohms  of  a  quadrant  cube 
of  the  material.  Hence  we  have 


How  the 
insulation 
resistance 
of  a  cable 
varies    with 
the  thick- 
ness of  the 
covering 


In  order  to  understand  how  the  insulation 
resistance  of  a  cable  depends  on  the  thickness 
of  the  insulating  covering  we  shall  find  the  in- 
sulation resistance  of  a  cable  consisting  of  a 
copper  cylinder  covered  with  a  given  thickness 
of  homogeneous  insulating  material  of  insulativity  o-.  We 
shall  suppose  that  the  conductor  is  at  potential  F,  and  that 
the  outside  of  the  covering  is  at  potential  zero.  This  would 
be  the  case,  for  instance,  if  the  cable  were  immersed  in  water 
contained  in  an  earthed  metal  tank.  The  stream  lines  of 
the  leakage  current  will  obviously  be  radial  to  the  cylinder. 

Let  us  imagine  that  the  cylin- 
der of  dielectric  is  divided  up  into 
an  infinite  number  of  thin  con- 
centric cylinders  (Fig.  9),  the  inner 
and  outer  radius  of  one  of  them 
being  x  and  x-\-dx  respectively. 
Consider  the  flow  in  a  centimetre 
length  of  the  conductor,  that  is, 
the  flow  from  the  inside  to  the 
outside  of  the  portion  of  the  di- 
electric contained  between  two  planes  each  perpendicular 
to  the  axis  of  the  cylinder  and  one  centimetre  apart. 
The  resistance  dBi  megohms  offered  to  this  flow  of  leak- 


FIG.  9. 


INSULATIVITY  51 

age  current  by  the    elementary  tube  of    dielectric    equals 

o-dx/2irx,  and  hence, 


where  rt  is  the  radius  of  the  copper  cylinder,  and  r2  is  the 
outer  radius  of  the  insulating  covering.  If  R  be  the  insula- 
tion resistance  of  a  length  I  cms.  of  the  conductor,  we  have 


for  the  flow  of  current  across  I  cms.  is  obviously  I  times 
the  flow  across  1  centimetre,  and  its  resistance,  there- 
fore, is  only  the  Ith  part  of  the  resistance  of  1  centi- 
metre. 

We  see  from  the  formula  that  if  n  is  to  be  kept  constant 
and  we  wish  to  increase  the  insulation  resistance  n  times, 
the  new  outer  radius  of  the  dielectric  would  have  to  be  equal 
to  ri  (r2/Vi)n,  and  hence  the  thickness  of  the  dielectric  would 
have  to  be  increased  from  r2  —  ri  to  (r2n  —  r^J/r^"1.  The 
ratio  of  the  new  thickness  to  the  old  would  therefore  be 
l+?*2  A*i+r22A"i2+-  .  -i-r2n~1/rin~l)  and  this  is  much  greater 
than  n,  except  when  r2/r±  is  nearly  equal  to  unity.  In  the 
same  way,  if  we  keep  r2  constant  and  diminish  r±  until  the 
insulation  resistance  is  n  times  as  great,  we  find  that  the 
ratio  of  the  new  thickness  to  the  old  is  1  -{-ri/r2-\-r12/r22-{- 
-\-rin~l/r2n~l)  and  this  is  much  smaller  than  n  ex- 
cept when  ri/ra  is  nearly  unity.  The  area  of  the  cross  section 
of  the  conductor  however  is  diminished  in  the  ratio  of  r^n 
to  r22n,  and  thus,  except  when  n  is  small,  and  r^  and  r2  are 
not  very  different,  it  will  be  exceedingly  small. 

The  above  results  illustrate  that  except  when  the  thickness 
of  the  insulating  covering  is  small  compared  with  the 
diameter  of  the  conductor,  increasing  the  thickness  of  the 
covering  is  not  an  economical  method  of  increasing  the 


52          ELECTRIC   CABLES   AND   NETWORKS 

insulation  resistance.  As  a  rule,  using  a  material  having  n 
times  the  insulativity  is  much  preferable  to  increasing  the 
thickness  of  the  insulation  n  times.  For  instance,  if  n  were 
4,  and  r2/ri  were  2,  the  insulation  resistance  in  the  former 
case  would  be  increased  four  times  whilst  in  the  latter  it 
would  be  increased  only  2-32  times. 

If  the  insulating  wrappings  round  a  wire  may  be 
regarded  as  concentric  cylinders,  each  cylinder  being  of 
homogeneous  material,  then  since  the  resistances  of  the 
cylinders  to  radial  flow  are  in  series,  the  resultant  insulation 
resistance  R  in  megohms  is  determined  by 

JR=((ri/27rZ)loge(r2/r1)+(a-2/27rZ)loge  (r3/r2)  +       . . 
where  <n,  0-2,     . .     are   the  insulativities,  and  r2,   r3, 
the  bounding  radii  of  the  various  wrappings.      This  formula 
shows  that  the  materials  ought  to  be  arranged  so   that  the 
values  of  <rly  a2,     . .     are  in   descending  order   of   magni- 
tude, the  material  having  the  greatest    insulativity  being 
next  the  conductor,  for  the  density  of  the  leakage  current 
is  a  maximum  next  the  conductor  and  diminishes  the  farther 
we  move  from  the  axis. 

The  insulativity  of  the  dielectric  varies  rapidly  with  the 
temperature,  but  unlike  the  resistivities  of  pure  metals,  it 
diminishes  as  the  temperature  increases.  Hence,  if  we 
assume  that  the  relation  follows  the  linear  law,  we  must 
write 

at=o-v{l—a(t—t')}. 

The  values  of  a  are  very  much  larger  than  for  metals. 
Thus  for  rubber,  Messrs.  Siemens  Bros.  &  Co.  give  0*047 
(Centigrade)  as  the  value  of  a,  and  for  gutta  0*16,  when 
Zis  15-6°  C.  (60°  F.). 

A.  Campbell  (Proc.  Roy.  Soc.  A.,  vol.  78,  p.  207)  gives  the 
following  table  to  show  how  the  insulativity  of  dry  cellulose 
varies  with  the  temperature. 


1NSULATIV1TY 


53 


Temp,  in  deg.  C. 

<T  x  10—  6 

25 

1,600 

30 

900 

40 

330 

50 

125 

60 

40 

65 


20 


measuring 
insulation 
resistance 


Hence  an  increase  of  40°  C.  causes  the  insulativity  to  dim- 
minish  to  one-eightieth  of  its  initial  value. 

As  the  accurate  measurement  of  the  insulation  resistance 
R  of  low  tension  cables  is  of  considerable  commercial 
importance  we  shall  describe  fully  the  method  which  by 
a  general  agreement  between 
manufacturers  and  engineers  is 
now  adopted  for  making  the  test. 

Method  of  Let  us  suppose  that 
the  insulation  resist- 
ance per  mile  of  a  coil 
of  110  yards  of  7/18  cable  has  to 
be  found.  The  coil  must  first  be 
immersed  in  water,  the  ends  being 
kept  dry,  at  60°  F.  for  24  hours 
previous  to  the  test.  The  ends  of 
the  coil  are  next  prepared.  The 
tape  and  protecting  material  is 
stripped  off  the  rubber  for  about 
6  inches  (Fig.  10)  from  the  ends 
of  the  cable.  The  rubber  is  then  stripped  from  the  con- 
ductor for  about  3  inches,  care  being  taken  that  the  por- 
tion of  the  rubber  left  on  is  intact. 


FIG. 


10. — The   ends   of   the 
cable  under  test. 


54 


ELECTRIC  CABLES  AND   NETWORKS 


B 


K 


Guard  Wire 


The  voltage  of  the  testing  battery  is  generally  taken  50 
per  cent,  greater  than  that  to  which  the  cable  will  be  sub- 
jected in  actual  working.  For  instance,  if  it  is  to  be  in- 
stalled in  a  building  supplied  from  direct  current  mains, 

having  220  volts  between 

G 


adjacent  mains  or  440 
volts  between  the  outers, 
the  testing  pressure  ought 
to  be  660  volts,  as  in 
practical  work  some  of 
the  wires  will  sometimes 
have  to  withstand  a  pres- 
sure of  440  volts  to  earth. 
The  battery,  galvano- 
meter, and  cable  are 
connected  in  series  as 
shown  in  Fig.  11.  When 
the  key  K  is  closed  there 
will  be  a  deflection  of  the 
ray  of  light  reflected  from 
the  mirror  of  the  galvano- 
meter, provided  that  the 
current  through  the  gal- 
vanometer is  sufficiently 
large.  The  current  leav- 
ing the  battery  flows 
through  the  water  and 
the  insulating  covering  of 
the  cable  to  the  copper 


FIG.  11. — Connexions  for  testing  the  in- 
sulation resistance  of  a  coil  of  cable. 


core,  and  then,  through  the  shunt  S  and  galvanometer  G  in 
parallel,  back  to  the  battery.  If  E  be  the  E.M.F.  of  the 
battery,  which  usually  consists  of  three  or  four  hundred 
small  accumulators,  we  have 


INBULATEV1TY  55 

E  S 


0  = 

where  B,  G,  S  and  R  are  the  resistances  of  the  battery,  gal- 
vanometer, shunt,  and  insulating  covering,  respectively,  and 
C  is  the  current  flowing  in  them.  Unless  the  cable  be 
broken  down  or  be  covered  with  very  inferior  insulating 
material,  B-\-GS/(G-\-S)  will  be  negligibly  small  compared 
with  R.  We  may  therefore  write  C  =  mE/R  or  R  —  mE/C, 
where  1/m  =  (G  -j-  S)/S  =  the  multiplying  power  of  the 
shunt. 

In  practice,  the  current  C  is  seldom  even  approxi- 
mately constant,  and  hence,  the  deflection  varies  with  the 
time  after  closing  the  switch.  As  a  rule  a  steady  deflection 
indicates  good  quality  material  and  a  very  unsteady  one 
that  the  insulativity  is  on  the  point  of  breaking  down.  For 
electric  lighting  cables,  the  convention  is  made  that  the 
deflection  is  read  after  one  minute's  electrification.  If  the 
galvanometer  be  calibrated,  we  know  the  value  of  C  corre- 
sponding to  a  given  deflection,  and  as  E  can  be  measured 
accurately  by  a  potentiometer  or  electrostatic  voltmeter,  R 
can  be  found.  Thus,  if  I  be  the  length  of  the  cable  in 
yards,  (//1760)jR  is  the  insulation  resistance  per  mile. 

To  calibrate  the  galvanometer  we  place  a  divided  megohm 
resistance  R,  and  an  accumulator,  in  series  with  it.  Let  us 
suppose  that  when  the  multiplying  power  of  the  shunt  is  1/Wi 
the  deflection  is  d\.  Then  the  current  m<JEj\/(B -\-m\Gr-\-R] 
gives  the  deflection  dlt  and  since  B  is  negligibly  small  com- 
pared with  R,  and  mlt  Eit  and  G,  can  be  accurately  de- 
termined, the  current  C  corresponding  to  a  deflection  di  can 
be  found.  The  voltage  E±  is  best  determined  by  comparing 
it  with  the  voltage  of  a  standard  cadmium  cell  by  a  poten- 
tiometer method.  The  E.M.F.  of  a  cadmium  cell  is  by 
Jaeger  and  Kahle's  formula, 


56         ELECTRIC  CABLES  AND  NETWORKS 

1-0186— 0-000038  (Z— 20)— 0-00000065(^—20) 2,  volts, 
at  t°  C.  By  varying  R  and  mt  other  deflections  can  be  found 
corresponding  to  known  currents.  Plotting  these  on  squared 
paper,  and  drawing  a  smooth  curve  through  the  points, 
we  get  an  accurate  calibration  curve  for  the  galvanometer, 
so  that  0  for  any  given  deflection  can  be  read  off  at  once. 

Price's  In    measuring    exceedingly    high    resistances 

guard 

wire         particular  care  has  to  be  taken  to  ensure  that 

We  are  not  merely  measuring  the  resistance  of  the  path  of 
some  surface  leakage  current.  From  Fig.  11,  we  see  that  a 
current  may  flow  from  the  water  along  the  surface  of  the 
insulating  covering,  and  then  pass  through  the  galvanometer 
without  passing  through  the  insulating  covering.  To 
obviate  this  source  of  error,  W.  A.  Price  uses  a  guard  wire  as 
shown  in  Figs.  10  and  11.  Apiece  of  "flexible"  does  ex- 
cellently for  the  purpose,  the  bare  end  of  the  flexible  being 
wrapped  round  the  rubber  insulation.  Practically  all  the 
surface  leakage  current  will  flow  along  the  guard  wire  with- 
out going  through  the  galvanometer  at  all.  In  this  case, 
therefore,  the  deflection  of  the  galvanometer  measures  only 
the  current  leaking  through  the  insulating  covering,  and 
hence,  we  can  find  the  true  value  of  the  insulation  resistance. 

The  grade         Cables  are  generally  divided  into   "  grades  " 

of 
insulation     according  to  their  insulation  resistance.     Cables 

belonging  to  the  600  megohm  grade,  for  instance,  have  an  in- 
sulation resistance  ranging  from  a  minimum  of  600  megohms 
per  mile  for  the  largest  sizes  up  to  2,000  megohms  per 
mile,  which  is  the  insulation  resistance  of  an  insulated  No. 
18  wire  belonging  to  this  grade.  Such  cables  are  made  of 
tinned  copper  conductors  insulated  with  pure  and  vulcanized 
rubber,  and  rubber  coated  tape,  the  whole  being  vulcanized 
together,  and  covered  with  braided  cotton  and  preservative 
compound.  The  list  price  of  this  grade  of  cable  ranges  from 


INSULAT1VITY  57 

about  £10  per  mile  for  1/22  wire  to  about  £1,100  per  mile 
for  61/12.  The  corresponding  cable  of  300  fl  (megohm) 
grade  would  be  about  5  per  cent,  cheaper,  and  for  2,500 
megohm  grade  cable  about  10  per  cent,  dearer. 
Institution  ^n  t^ie  w^ng  ru^s  (1907)  of  the  English  In- 
rules  stitution  of  Electrical  Engineers,  the  dielectrics 
for  cables  are  divided  into  two  classes.  In  the  first  or  A 
class,  the  dielectrics  of  the  cables  are  impervious  to  moisture 
and  only  need  mechanical  protection.  Vulcanized  rubber, 
for  instance,  would  belong  to  this  class.  In  the  B  class  are 
included  those  dielectrics  like  paper  or  fibre  which  must  be 
kept  perfectly  dry.  They  therefore  need  to  be  encased  in  a 
waterproof  sheath.  This  generally  consists  of  a  soft  metal 
tube,  a  lead  tube  for  example,  drawn  closely  over  the 
dielectric. 

In  the  following  table  the  minimum  insula- 

Minimum 

insulation  tion  resistances  of  vulcanized  rubber  (class  A) 
cables,  in  megohms  per  mile,  approved  by  the 
Institution  of  Electrical  Engineers  are  given.  These  in- 
sulation resistances  are  those  of  cables  of  I.E.E.  600  and 
2,500  megohm  grades  respectively.  The  minimum  insula- 
tion resistances,  advisable  in  practice,  when  the  dielectric  is 
fibre  or  paper,  lead  covered,  are  also  given.  It  has  been 
considered  advisable  to  fix  minimum  values  to  the  radial 
thicknesses  of  the  dielectric  used  in  different  sized  and 
different  grade  cables.  This  latter  procedure,  however,  is 
open  to  criticism  as  the  mechanical  strength  of  the  different 
kinds  of  dielectrics  used  in  practice  vary  largely  and  the 
restriction  tends  to  put  them  all  on  the  same  level.  The 
minimum  radial  thicknesses  are  quoted  in  the  table. 

The  dielectric  used  must  not  soften  at  temperatures 
lower  than  176°  Fahr.  (80°  C.)  as  otherwise  there  would  be  a 
risk  of  the  conductor  gradually  sinking  in  it  and  ultimately 


58 


ELECTRIC  CABLES  AND  NETWORKS 


touching  the  sheath.  It  is  customary  to  apply  an  alternat- 
ing pressure  of  2,000  volts  for  half  an  hour  between  the 
conductor  and  the  sheath,  after  the  cable  has  been  immersed 
in  water  for  24  hours,  as  this  pressure  will  probably  break 
down  any  weak  part  in  the  dielectric  covering.  The  wave  of 
the  applied  P.D.  must  be  sine  shaped  and  the  frequency 
of  the  alternating  current  should  be  50. 


TABLE 


Gauge 

Minimum  Insulation  Resist- 
ance in  megohms  per  mile 

Minimum  Radial  Thickness 
in  mils. 

Vulcanized  rubber 

Vulcanized  Rubber 

Diameters  of 
strands  are  given 

Class  A 

Fibre 
or  Paper 
Class  B 

Class  A 

Fibre 
or  Paper 
Class  B 

in  S.W.G.  or  ins. 

Up  to 
250  volts 

Up  to 
650  volts 

Up  to 
250  volts 

Up  to 
650  volts 

3/25 

2,000 

5,000 

140 

34 

62 



3/24 

2,000 

5,000 

140 

34 

62 

— 

3/23 

2,000 

5,000 

140 

35 

62 

— 

1/18 

2,000 

5,000 

140 

35 

62 

— 

3/22 

2,000 

5,000 

140 

36 

62 

— 

7/25 

2,000 

5,000 

140 

36 

62 

— 

3/21 

2,000 

5,000 

140 

38 

62 

— 

1/17 

2,000 

5,000 

140 

36 

62 

— 

7/24 

2,000 

5,000 

140 

37 

62 

— 

3/20 

2,000 

5,000 

140 

38 

62 

— 

7/23 

2,000 

5,000 

140 

37 

62 

— 

1/16 

2,000 

5,000 

140 

36 

62 

— 

3/19 

1,250 

4,500 

140 

39 

62 

— 

1/15 

1,250 

4,500 

140 

37 

62 

— 

7/22 

1,250 

4,500 

140 

39 

62 

— 

1/14 

1,250 

4,500 

140 

38 

62 

— 

3/18 

1,250 

4,500 

140 

40 

62 

— 

7/21 

1,250 

4,500 

140 

40 

62 

— 

7/20 

900 

4,000 

140 

41 

62 

— 

7/19 

900 

4,000 

140 

42 

•62 

— 

7/18 

900 

4,000 

140 

44 

62 

80 

7/17 

900 

4,000 

140 

47 

62 

80 

INSULATIV1TY 


59 


Gauge 

Minimum  Insulation  Resist-         Minimum  Radial  Thickness 
ance  in  megohms  per  mile                              in  mils. 

Vulcanized  Rubber 

Vulcanized  Rubber 

Class  A 

Fibre 

Class  A 

Fibre 

Diameters  of 
strands  are  given 

r  Paper 
Class  B 

or  Paper 
Class  B 

inS.W.G.  or  ins. 

Up  to 

Up  to 

Up  to         Up  to 

250  volts 

650  volts 

250  volts  650  volts 

19/20 

900 

3,500 

140 

48 

62 

80 

7/16 

900 

3,500 

140 

49 

62 

80 

19/19 

750 

3,500 

140 

50 

62 

80 

7/0-068  in. 

750 

3,500 

140 

51 

62 

80 

7/15 

750 

3,500 

140 

52 

62 

80 

19/18 

750 

3,000 

120 

54 

62 

80 

7/14 

750 

3,000 

120 

54 

62 

80 

19/17 

750 

3,000 

120 

58 

62 

80 

7/0-095  in. 

750 

3,000 

120 

59 

62 

80 

19/0-058  in. 

750 

3,000 

120 

59 

62 

80 

19/16 

750 

3,000 

110 

62 

66 

80 

19/15 

600 

3,000 

110 

66 

66 

80 

19/14 

600 

3,000 

100 

71 

71 

90 

19/0-082  in. 

600 

3,000 

100 

71 

71 

90 

37/16 

600 

3,000 

90 

76 

76 

90 

19/13 

600 

3,000 

90 

76 

76 

90 

37/15 

600 

3,000 

90 

80 

80 

90 

19/0-101  in. 

600 

3,000 

90 

80 

80 

90 

37/14 

600 

2,500 

90 

87 

87 

90 

37/0-082  in. 

600 

2,500 

90 

87 

87 

90 

37/0-092  in. 

600 

2,500 

80 

94 

94 

100 

37/0-101  in. 

600 

2,500 

80 

101 

101 

100 

37/0-110  in. 

600 

2,500 

80 

107 

107 

100 

61/13 

600 

2,500 

80 

113 

113 

100 

61/0-098  in 

600 

2,500 

80 

121 

121 

100 

61/0-101  in. 

600 

2,500 

80 

121 

121 

100 

61/0-108  in 

600         2,500 

80 

125 

125 

110 

61/0-110  in 

600 

2,500 

80 

125 

125 

110 

61/0-118  in 

600 

2,500 

80 

129 

129 

110 

91/0-098  in 

600     !    2,500 

70 

129 

129 

110 

91/0-101  in 

600 

2,500 

70 

133 

133 

110 

91/12 

600 

2,500 

70 

133 

133 

120 

91/0-110  in 

600 

2,500 

70 

137 

137 

120 

91/0-118  in 

600 

2,500 

70          141 

141 

130 

127/0-101  in 

600         2,500 

70 

141          141 

130 

60         ELECTRIC  CABLES  AND  NETWORKS 

In  the  above  table  the  insulation  resistance  R±  is  given  in 
megohms  per  mile.  This  is  the  unit  customarily  employed 
in  England.  On  the  Continent,  the  insulation  resistance  R 
is  generally  given  in  megohms  per  kilometre.  As  a  mile 
equals  1-609  kilometres,  it  follows  that  ^  =  1-609^. 

Methods  of        In  proving  the  formulae  for  insulation  resist- 
measuring 

a-  ance   given  above,  it  must  be  noticed  that  we 

have  made  the  assumption  that  the  insulating  material  is  of 
homogeneous  substance.  In  finding  the  value  of  a,  there- 
fore, care  has  to  be  taken  that  the  sample  experimented  on 
is  homogeneous.  When  insulating  materials  are  obtained 
in  thin  sheets  for  testing  purposes,  the  thin  sheets  are  often 
varnished.  As  this  varnish  has  usually  a  higher  insulati- 
vity  than  the  material  in  the  interior  of  the  sheets,  we  should 
expect  that  the  values  of  o  found  by  tests  on  thin  sheets 
would  be  greater  than  the  values  found  by  tests  on  thick 
sheets,  and  this  is  found  to  be  the  case.  When  also  the 
dielectric  sheets  are  laid  between  metal  plates,  with 
weights  placed  on  the  upper  one,  the  value  of  the  resistance 
between  the  sheets  is  found  to  vary«  with  the  mechanical 
pressure  and  with  the  testing  voltage,  the  resistance  being 
smaller  the  greater  the  pressure  and  the  greater  the  voltage. 
R.  Appleyard  has  shown  that  it  is  possible  to  get  consistent 
results  by  testing  the  sheets  between  suitable  mercury 
electrodes.  His  method  of  testing  is  as  follows.  The  sheet 
of  dielectric  is  placed  vertically  between  two  flat  rings  of 
ebonite  faced  on  each  other  with  soft  india  rubber.  Disks 
of  iron  are  clamped  over  each  ring  and  form  the  jaws  of  a 
large  ebonite  vice.  Mercury  is  poured  into  the  hollow  spaces 
between  the  iron  rings  and  the  dielectric,  through  holes  on 
the  top  of  each  disk.  The  temperature  can  be  conveni- 
ently read  by  placing  a  thermometer  in  the  mercury.  With 
this  arrangement  he  found,  by  experiments  on  presspahn, 


INSULATIVITY  61 

that  the  dielectric  resistance  is  sensibly  the  same  whatever 
the  testing  voltage  may  be,  and  that  it  is  practically  in- 
dependent of  the  time  of  application  of  the  pressure. 

In  the  following  table  rough  approximations  to  the  value 
of  0-  for  various  insulating  materials  are  given.  As  the  in- 
sulativity  a  generally  varies  very  rapidly  with  temperature, 
the  numbers  quoted  only  indicate  the  order  of  the  magnitude 
of  a-  which  might  reasonably  be  expected. 


Dielectrics 

a-  xlO—  6 

Mica  

84 

Gutta       

450 

Rubber    

10,000 

Ebonite          

30000 

Glass  

20,000 

When  two  or  three  hundred  yards  of  a  cylindrical  wire 
cable  insulated  with  a  known  thickness  of  the  insulating 
material  is  available,  we  can  find  a  by  measuring  the 
insulation  resistance  of  the  cable. 

For  instance,  if  R±  is  the  insulation  resistance  of  the  cable 
in  megohms  per  mile,  we  have,  with  our  usual  notation 


where  I  is  the  number  (160,900)  of  centimetres  in  a  mile. 

Hence 

_27r(160,900)  x  0-43437?! 


=439  000 
=272  900  tf/logu  (ra/rO, 

where  E  is  the  insulation  resistance  in  megohms  per  kilo- 
metre. We  can  thus  by  finding  Rv  or  E,  and  ri  and  r2 
determine  a-. 

Numerical         Example    I.      In     the    Ferranti     concentric 
examples      mauij  wnich  connected  the  generating  station  at 


62          ELECTRIC   CABLES   AND   NETWORKS 

Deptford  with  the  distributing  station  at  Trafalgar  Square, 
the  insulating  material  consisted  of  brown  paper  and  black 
wax.  The  insulation  resistance  per  mile  after  it  was  laid 
was  found  to  be  720  megohms.  The  outer  radius  of  the  inner 
conductor  was  0-406  inches,  and  the  inner  radius  of  the  outer 
conductor  was  0-922  inches.  Hence,  by  our  formula 
o-=439  000  x720/log( 922/406) 

=887-6  xlO6 
It  is  therefore  better  than  gutta  but  inferior  to  rubber. 

Example  II.  The  insulation  resistance  of  a  mile  of  cable 
is  1,000  megohms,  the  radius  of  the  copper  is  0-4  of  an 
inch,  and  of  the  insulating  covering  0-97  of  an  inch.  What 
is  the  average  value  of  the  insulativity  ? 

We  have,  cr=439  000  x  1  000/log( 97/40) 

=  1  140  xlO6. 

Example  III.  If  the  insulativity  be  10  x  106,  and  the 
ratio  of  the  outer  to  the  inner  radius  of  the  insulating  cover- 
ing of  a  main  be  2,  find  the  insulation  resistance  of  the  main 
in  megohms  per  kilometre. 

In  this  case  #=(107/272-9)log2 

=  1  100  megohms  nearly. 

Example  IV.  If  the  inner  and  outer  radii  of  the  insul- 
ating covering  of  a  cable  with  a  cylindrical  core  be  0-2  and 
0-3  cm.  respectively,  and  its  insulation  resistance  is  1000 
megohms  per  mile,  what  would  be  the  insulation  resistance 
of  a  cable  consisting  of  a  copper  cylinder  0-5  cms.  radius 
covered  with  insulating  material  to  a  depth  of  0-1  cm.  ? 
From  the  data  given  for  the  first  cable,  we  have 

1  000=(<r/439  000)log(l-5) 
and  for  the  second, 

#=(oy439  000)log(l-2), 
and  therefore,  E=  |log(l-2)/log(  1-5) }  1000 
=449-8. 


INSULATIVITY  63 

The  best  material  for  insulating  cables  is 
rubber.  It  is  a  vegetable  product  being  the 
coagulated  milky  juice  of  various  trees  and  shrubs.  It 
contains  a  small  amount  of  resinous  matter  soluble  in 
alcohol.  The  rubbers  obtained  from  Para,  Ceara,  and  Mada- 
gascar which  contain  very  little  resinous  material  are  the 
most  expensive,  and  those  from  Guatemala  and  Africa 
which  contain  much  more  resinous  material  and  are  not  so 
suitable  for  insulating  purposes  are  cheaper.  Para  rubber, 
which  is  generally  considered  the  best,  is  obtained  from  a 
large  euphorbiaceous  tree  about  50  feet  high.  The  exact 
chemical  formula  for  rubber  is  not  yet  known,  but  carbon 
and  hydrogen  are  its  only  constituents.  Its  specific  gravity 
is  about  0-92.  It  is  very  hygroscopic,  the  weight  of  the 
moisture  absorbed  being  about  20  per  cent,  of  its  own 
weight.  At  0°  C.  it  is  rigid  and  cannot  be  easily  elongated, 
but  it  is  not  brittle.  When  heated  to  temperatures  less 
than  100°  C.  it  becomes  soft  and  easily  stretched,  but  at 
120°  C.  it  practically  loses  its  power  of  recovery  when 
stretched.  At  200°  C.  it  is  a  thick  viscous  liquid,  and  when 
heated  still  more  it  is  converted  into  hydrocarbons  and 
only  a  small  carbonaceous  ash  is  left. 

If  rubber  be  exposed  to  the  effects  of  atmospheric  changes 
it  oxidizes  and  deteriorates  rapidly.  For  this  reason  it  is 
hardened  and  vulcanized  by  the  action  of  sulphur.  About 
3  per  cent,  of  sulphur  is  mixed  with  the  rubber  in  the 
vulcanizing  chamber  at  160°  C.  The  sulphur  combines 
chemically  with  the  rubber  forming  a  fairly  soft  and  elastic 
material.  After  being  vulcanized  (cured)  the  elasticity  of 
the  rubber  is  greatly  increased  and  it  is  not  hardened  by 
cold  or  softened  by  heat. 

Ozone  attacks  and  destroys  rubber  rapidly,  hence,  if  brush 
discharges  are  likely  to  take  place,  it  must  not  be  used  in 
high  tension  cables.  Grease  has  a  deleterious  action  on 


64          ELECTRIC   CABLES   AND   NETWORKS 

rubber.  It  darkens  its  colour  and  makes  it  sticky.  When 
a  larger  percentage  of  sulphur  is  mixed  with  the  rubber,  and 
it  is  subjected  for  a  longer  time  to  the  action  of  heat,  we 
get  ebonite  (vulcanite). 

If  there  is  an  appreciable  quantity  of  free  sulphur  in  the 
vulcanized  rubber  it  will  attack  the  copper  conductor. 
For  this  reason  it  is  customary  to  tin  the  copper  wires  used 
in  cables  (see  p.  29). 

Gutta  like  rubber  is  the  dried  milky  juice  of 
various  trees.  The  most  important  is  the 
Isonandra  Gutta,  of  the  order  Sapotaceae,  found  in  the  Ma- 
layan Archipelago.  Unlike  rubber  it  is  practically  inelastic, 
and  as  it  softens  at  a  low  temperature  it  is  little  used  for 
insulating  electric  light  cables.  When  insulating  wires 
with  gutta,  they  are  first  passed  through  a  bath  of  Chat- 
terton's  compound,  and  are  then  passed  through  a  press 
heated  so  that  the  gutta  is  in  the  liquid  state.  The  gutta  is 
forced  out  round  the  wire  as  it  leaves  the  die.  It  next  passes 
through  cold  water  to  stiffen  it.  A  second  or  third  coating 
can  then  be  put  on  in  a  similar  manner. 

REFERENCES. 

S.  A.   Russell,  Electric  Light  Cables. 
C.    Baur,   Das   Elektrische   Kabel. 

F.  A.  C.  Perrine,  Conductors  for  Electrical  Distribution. 
Sir  William  Preece,  "  On  the  Specification  of  Insulated  Conductors 

for  Electric  Lighting  and  other  Purposes."     Journ.  of  the  Inst. 

of  Elect.  Engin.,  vol.   xx.,   p.    605,    1891. 
R.  Appleyard,  "  Contact  with  Dielectrics."     Phil.  Mag.  [6]  vol.  10, 

p.  485,  1905. 
E.  H.  Rayner,  "  Temperature  Experiments."     Journ.  of  the  Inst. 

of  Elect.  Engin.       Vol.  xxxiv.,  p.  613,  1905. 
J.   Langan,    **  Standardizing  Rubber  covered  Wires   and   Cables." 

Proc.  Am.  Inst.  Elect.  Engin.,  vol.  25,  p.  189,  1906. 
W.  S.  Clark,  "  Comments  on  Present  Underground  Cable  Practice." 

Proc.  Am.  Inst.  Elect.  Engin.,  vol.  25,  p.  203,  1906. 
A.   Schwartz,  "  '  Flexibles,'  with  Notes  on  the  Testing  of  Rubber." 

Journ.  of  the  Inst.  of  Elect.  Engin.,  vol.  xxxix.,  p.  31,  1907.     A 

useful  bibliography  is  given  at  the  end  of  this  paper, 


DISTRIBUTING  NETWORKS 


CHAPTER    IV 

Distributing  Networks 

Kelvin's  law — Distributing  networks — Copper  mains — Distributing 
centre — Example — The  economy  of  high  pressure — Uniformly 
distributed  load — Excessive  current  density — Main  with  a 
branch  circuit— Sections  of  the  mains  all  different— Numerical 
example — Feeding  from  both  ends — Single  tapping — Two 
centres  feeding  a  distributing  centre  and  a  branch  main — Loop 
fed  from  one  centre — Loop  with  several  feeding  centres — Ring 
main  with  n  feeding  points — The  proper  site  of  the  power  sta- 
tion— Example — The  feeding  centre  for  a  straight  main — 
Practical  rule— Booster— The  economy  of  a  booster— References. 

Kelvin's  ASSUMING   that   the   generating   voltage   has 

law  been  fixed,  let  us  consider  the  problem  of  trans- 
mitting a  definite  amount  of  electrical  energy  by  means 
of  a  direct  current  of  given  magnitude  C  from  one  station 
to  another.  If  R  be  the  total  resistance  of  the  mains  for 
the  outgoing  and  the  return  current,  the  power  expended 
in  heating  them  will  be  C2R,  and  since  R  is  inversely  pro- 
portional to  the  area  x  of  the  cross  section  of  the  main 
used,  we  see  that  the  annual  cost  of  the  energy  expended 
in  heating  the  conductors  may  be  written  in  the  form  /JL/X, 
where  //.  is  a  constant  depending  on  the  cost  at  which  power 
can  be  generated,  the  character  of  the  load,  etc.  By  increas- 
ing x  we  diminish  the  annual  cost  of  the  power  that  would 
be  expended  in  heating  the  mains,  but  we  increase  the 
initial  cost  of  the  mains,  and  therefore,  the  annual  sum  that 
has  to  be  expended  in  interest  on  the  capital  borrowed,  and 

67 


68          ELECTRIC  CABLES   AND  NETWORKS 

laid  aside  for  the  depreciation  in  the  value  of  the  mains. 
It  is  obvious,  therefore,  that  the  most  economical  con- 
ductor to  choose  is  the  one  for  which  the  annual  interest 
and  depreciation  on  the  initial  cost  together  with  the  annual 
cost  of  the  energy  wasted  is  a  minimum.  In  practice, 
therefore,  we  should  have  to  find  the  sum  of  these  annual 
charges  for  cables  of  the  various  sizes  given  in  manu- 
facturers' catalogues  and  choose  the  cable  for  which  this 
sum  is  a  minimum. 

In  the  particular  case  where  the  interest  and  depreci- 
ation on  the  initial  cost  of  the  cable  is  proportional  to  the 
weight  of  the  conducting  material  used,  and  therefore 
proportional  to  x,  the  section  of  the  required  cable  can  be 
determined  very  simply  mathematically.  In  this  case  the 
interest  and  depreciation  may  be  expressed  by  \x  where 
X  is  independent  of  x.  Hence  the  total  annual  charge 
for  the  mains  will  be  Xx+p/x,  that  is,  {(\x)l/2—(fjL/x)1/z}z 
-f-2(Xyu,)1/2.  Since  the  least  possible  value  of  the  square 
of  a  number  is  zero,  and  2(X/n)1/2  is  independent  of  x,  we  see 
that  the  total  annual  charge  is  a  minimum  when  ^x=/j,/x. 
We  thus  deduce  that  the  most  economical  conductor  to 
use  is  that  for  which  the  interest  and  depreciation  on  its 
initial  cost  equals  the  annual  cost  of  the  energy  expended  in 
heating  it.  This  is  generally  known  as  Kelvin's  law.  If  the 
initial  cost  of  the  cable  be  only  approximately  proportional  to 
the  area  of  the  cross  section,  the  rule  still  gives  a  useful  indi- 
cation of  the  probable  size  of  the  most  economical  conductor. 
In  England  the  permissible  voltage  drop  in 
ing  the  mains  is  determined  by  the  Board  of  Trade 
Regulations.  The  seventh  rule  (B.  7)  reads 
as  follows  :  — 

"  7.  Variation  of  Pressure  at  Consumer's  Terminals. — • 
The  variation  of  pressure  at  any  consumer's  terminals 


DISTRIBUTING  NETWORKS  69 

shall  not  under  any  conditions  of  the  supply  which  the 
consumer  is  entitled  to  receive,  exceed  4  per  cent,  from 
the  declared  constant  pressure." 

To  illustrate  how  this  limitation  affects  the  design  of  a 
network,  let  us  first  consider  the  case  of  a  2  wire  dis- 
tributing system  (Fig.  12).  At  the  points  AI,  A2)  A3,  .  .  . 

Ai    At  As    A4  Ag-  A-*. 

lYl  • 


,       ,      , 
A,    A,'  A,    A4  A*  A'n 

FIG.   12. 

let  the  currents  (7l3  C2  ,03,  ...  be  required,  and  let  the 
distances  MAlt  MA2,  .  .  .  from  the  station  terminals 
be  denoted  by  Z1?  12,  .  .  .  We  shall  suppose  that  the 
section  of  the  mains  is  uniform  throughout  and  that  the 
potential  differences  between  the  points  MM',  A^A^, 
A2A2',  ...  are  V,  Fi,  F2,  .  .  .  The  portions  of 
the  mains  MA±  and  M'A±,  are  traversed  by  a  current 
01+02+.  .  .+0n.  We  have,  therefore, 

F—  F1=(01+0a  +  .    .    .+Cn)  (2ph/8), 

where  p  is  the  volume  resistivity,  and  8  the  area  of  the  cross 
section  of  the  mains.  Similarly  we  find  that 


and  F^  -Vn=Cn  {  2P(ln-ln_1)/S  }  . 

Thus  the  potential  difference  drop  p  at  the  most  distant 
feeding  point  An  is  given  by 

P  =  V—  Vn=(2p/8)(CJ1+C2l>  +  .    .    .+0BZW) 
=(2P/8)SCl       ..............     (1). 

If  I    denote    the    distance    from    M    of    the     centre   of 


70 


ELECTRIC  CABLES  AND  NETWORKS 


parallel  forces  equal  to  C^  C2,   .    .    .   acting  at  the  points 
AI,  A  2)     .    .    .   when  the  mains  are  straight,  we  have 


and  thus,  from  (1), 

8=(2p/pjlSC  ......     (2). 

Hence,  when  p  is  given,  (2)  determines  the    cross  section 

of  the  main. 

Copper  -^or  c°PPer  mains,  when  the  temperature  is 

mains     1Qo  Q  ^  ^  =  1665  xlO-9ohms,  and  thus  (2)  becomes 


If  /  be   measured  in  metres,   and  S  in  square    mms.,  we 
get 


=(T/30p)5C       ........     (3), 

approximately.     This   formula   is    often   used    by   French 
electricians  and  is  convenient  in  practice. 

When  making  calculations  instead  of  showing  both 
mains  in  the  diagram  it  is  sufficient  to  show  one  only 
(Fig.  13),  since,  in  practice,  we  may  regard  the  return  main 
as  identical  with  the  outgoing  main. 


M*. 


A   A. 


c,    c. 


13. 


Distri- 
buting 
centre 


If  G  be  the  point  in  the  main  MAn,  at  which, 
if  all  the  current  2C  were  taken,  the  voltage 
drop  between  M  and  An  would  be  the  same  as 
in  the  actual  case,  G  is  called  the  distributing  centre  of 
the  load.  If  we  suppose  that  the  main  is  stretched  straight 


DISTRIBUTING  NETWORKS  71 

and  that  weights  equal  to  Ci,  C2,  .  .  .  Cn  are  placed  at 
A1}  A 2,  .  .  .  An,  respectively,  G  will  be  the  centre  of 
gravity  of  these  weights.  Hence  we  can  use  the  ordinary 
statical  formula 

lSC=CJi+CJa+.    .    .+Cnln, 
to  determine  the  length  I  of  MG. 

Let   us   suppose   that   there   are    five   distri- 
Example 

buting  points  each  25  metres  apart,  and  that 
the  distance  of  the  first  distributing  point  from  the  station 
is  50  metres.  Let  the  currents  required  at  Al9  A2,.  .  . 
A6,  be  5,  10,  30,  10  and  5  respectively.  A3  is  obviously 
the  centre  of  gravity  and  thus  1  =  100.  Hence  substitut- 
ing in  (3),  we  find  that 

8  =  100  x60/30p=200/p  sq.  mms. 
The  From  (3)  we  have 

economy  

of  high  p=(l/ZOS)2C      (4). 

pressure 

If  we  increase  the  pressure  of  supply  n  times  the 
permissible  value  of  p  is  generally  increased  n  times  also, 
as  the  Board  of  Trade  rule  fixes  the  percentage  variation 
of  the  pressure,  and  not  its  absolute  magnitude.  We 
see,  therefore,  from  (4)  that  with  the  same  mains  we  can 
supply  n  times  the  current.  But  we  have  also  increased 
the  pressure  n  times,  and  hence  the  load  we  can  supply 
with  the  same  mains  is  increased  n2  times.  If,  for  example, 
we  increase  the  pressure  from  100  to  250  volts,  we  can 
increase  the  maximum  permissible  load  (250/100)2,  that 
is,  6-25  times.  For  this  reason  it  is  economical  to  supply 
at  the  highest  permissible  pressure. 

Let  us   now  suppose  that  the  currents   are 

Uniformly 

distributed     taken  from  points  A l5  A 2,  Az,   ,    .    .   An  (Fig.  14) 
at   equal   distances   apart   and   that   the   main 
is  fed  from  M .     Let  us  also  suppose  that  the  currents  are 
all  equal  to  c,  that  MA^=a,  and  that  AiA2=A2A3  =  .    .    . 


72          ELECTRIC  CABLES  AND  NETWORKS 

=x.  The  distributing  centre  of  the  currents  is  at  a  dis- 
tance (n — l)x/2  from  Alt  and  thus^=  MG=a-\-(n — I)x/2, 
and  2c=nc=C. 


M, 


FIG.  14. 


Hence,  by  (2), 

p=(2P/S){a+(n—  l)x/2}C 
=(2p/S)[a/2+  {  a+(n-l)x  }  /2]C 
=(pa/S)C+EC 


where  R  and  E±  are  the  resistances  of  the  whole  main  M  An, 
and  the  part  MAt  respectively.  It  follows  that  if  the 
load  be  uniformly  distributed  along  the  main  from  A± 
to  A  we  have 


n, 


If  the  load  had  been  concentrated  at  An,  p  would  equal 
2EC.  Hence,  except  in  the  case  when  MA^  is  negligibly 
small,  the  voltage  drop  with  a  uniformly  distributed  load 
is  slightly  more  than  half  the  value  it  has  when  the  load 
is  concentrated  at  the  far  end. 

If  H  be  the  power  in  watts    expended  in  heating    the 
mains,  then  in  the  case  represented  in  Fig.  14,  we  have 
H/2=(pa/S)C*+(px/S){(n—  l)2+(n—  2)2+-  .  .+ 

{2—  l/n}C2/6 


For  a  load  uniformly  distributed  n  is  infinite,  and  thus 

H/2=(2/3)E1C2-\-(l/^)EC2. 
If  the  load  had  been  concentrated  at  the  far  end  of  the 


DISTRIBUTING  NETWORKS  73 

line,  the  value  of  H/2  would  have  been  RC2,  and  there- 
fore, if  MA±  be  small,  the  power  expended  in  heating 
the  mains  when  the  load  is  uniformly  distributed  is  very 
little  more  than  one-third  of  its  value  with  all  the  load  at 
the  far  end. 

Let  us  now  consider  the  case  of  a  main  ML 

from  both     (Fig.   15)   uniformly  loaded  and  supplied  from 
ends 

both   ends.     If   I   be   the   length   of   the   main 

and  C  be  the  total  current  required,  (7/2  will  be  the 
current  flowing  in  at  each  end,  and  the  greatest  permissible 
voltage  drop  p  will  be  at  the  middle  of  the  main.  Hence 
p=(R/2)  ((7/2)  where  R  is  the  resistance  of  the  whole  main, 


M.. 


A3 


A., 


A,.,       A. 


-.  L 


FIG.  15. 

and  thus  RC=4=p.  If  the  main  had  been  supplied  from  one 
end  only,  the  greatest  value  C'  of  the  current  would  be 
given  by  RC'  =p.  For  the  same  maximum  voltage  drop, 
therefore,  we  could  supply  four  times  as  much  current 
when  we  feed  from  both  ends  of  the  main,  but  the  losses 
in  heating  the  mains  would  be  sixteen  times  greater  in 
the  latter  case. 

It  sometimes  happens  that  the  cross  section 
current       of  the  main  found  by  formula  (2)  makes  the 
current    density    too    high.     In    this    case,  the 
greatest  permissible   current   density  is   chosen.     In  only 
a  few  cases  would  it  be    advisable    to    choose  a  current 
density  as   high    as    2'5    amperes   per   sq.    mm.    (approxi- 
mately    1,600     amperes     per     sq.     in.).      Suppose,     for 


74 


ELECTRIC  CABLES  AND  NETWORKS 


instance,  that  p  is  2,  1=20  metres  and  %C=30  (Fig.  16). 
Formula  (2)  gives 

£=20x30/(30x2)=10  sq.  mms. 


M 


Zo 


Ao 

FIG.  16. 

This  would  give  a  current  density  of  30/10,  that  is,  3 
amperes  per  sq.  mm.  It  would  be  better,  therefore,  to 
make  the  area  of  the  cross  section  15  sq.  mms.  so  as  to 
reduce  the  current  density  to  30/15,  that  is,  2  amperes 
per  sq.  mm. 

Main  Let  us  suppose  that  the  main  Mab  (Fig.  17) 

^as  a  branch  ca  joining  it  at  a.     Let  us  also 
suppose  that  Mab  is  the  main  circuit  so  that 
the  section  of  Mab  is  uniform.     Let  Ci,  c2,   .    .    .   be  the 


branched 
circuit 


M 


c 
FIG.   17. 


currents  tapped  off  between  M  and  a,  at  distances  di,  d2, 
.    .    .  from  M .     Let  c/,  c2',    .    .    .   be  the  currents  tapped 


DISTRIBUTING  NETWORKS  75 

off  between  a  and  b,  at  distances  dit  d2',  .  .  .  from  a,  and 
let  Ci",  c2",  ...  be  the  currents  taken  between  a 
and  c.  We  see,  by  (4),  that  the  drop  of  voltage  between 
a  and  b  is  (a<72/30$)3V,  where  g2  is  the  distributing  centre 
of  the  currents  c/,  c/,  etc.  Similarly  if  gv  and  g3  be  the 
distributing  centres  of  the  currents  Ci,  c2,  .  .  .  and  C/' 
c/,  .  .  .  respectively,  the  voltage  drop  p  between  M 
and  b  is  given  by 

p  =  {Mgl.C+Ma.(C'+C")+ag2.C'}/WS, 
where  C,  C'  and  C"'  stand  for  Jc,  3c'  and  £c"  respectively. 
If  we  write  d  for  Mg1}  d'  for  agr2,  and  /  for  Ma,  this  formula 
becomes 

p  =  {  dC+l(C'+C")  +d'C'  }  /30tf, 
and  hence, 

S  =  {dC+l(C'+C")+d'C'}/30p    ..      ..     (5). 
The  voltage  drop  p^  from  M  to  a  is  given  by 


and  the  voltage  drop  p2  from  a  to  c  by 


where  d"  equals  ag3.     Hence,  if  p^-\-p2—py  we  must  have 
p2=p  —  pi9  and  therefore, 

S"=d"C"/W(p—p,)=S(d"C"/d'C')  .  .  (6). 
Hence  if  d"C"  be  greater  than  d'C',  S"  will  be  greater 
than  S. 

We  have  now  to  consider  whether  it  would  be  more 
economical  to  make  the  section  of  ab  or  the  section  of  ac 
the  same  as  that  of  Ma.  Let  F6  denote  the  volume  of 
the  copper  required  in  the  first  case,  and  Vc  the  volume 
required  in  the  second.  If  the  lengths  of  ab  and  ac  are  V 
and  I"  respectively,  we  have 

F5/2  =8(1+1')  +S(d"C"/d'C')l" 

=[  {  dC+l(C'+C")  +d'C'  }  /30p]  {l+l'+l"(d"C"/d'C')  }  . 
We  also  have 


76         ELECTRIC  CABLES  AND  NETWORKS 


Vc/2  =[  {  dC+l(C'  +  C"}  +d"C" 
and  hence, 

Vb—Vc=2{  (d'C'—d"C")/mp  }  [l—(l'/d"C" 


If  therefore  (d'C'  —  d"C")  and  [I—  (lf  /d"C"+l"  /d'C') 
(dC-\-l(Cf-\-Cff)}'\  have  the  same  sign,  Vb  is  greater  than 
Vc  and  thus  ac  should  be  made  the  principal  branch.  If 
they  have  not  the  same  sign,  ab  should  be  made  the 
principal  branch. 

Sections          -^e^  us  now  suPPose  that  the  sections  of    the 
mainsail      three  mains  Ma,  ab  and  ac   (Fig.    17)   are   all 

different  different  but  that  there  is  the  same  voltage 
drop  between  M  and  b  and  between  M  and  c.  Let  V 
be  the  volume  of  the  copper  used  and  x  the  voltage  drop 
to  a.  Then  using  the  same  notation  as  in  the  last  section 
we  have 

V/2  =  {dC+l(C'+C")  }l/3Qx+d'C'l'/30(p—  x) 

+d"CT/M(p—x)  =A/x+B(p—x), 
where 

A=  {dC+l(C'+C")}l/W,  and  B  =  {dfC'l'+d"CT}  /30. 
By  the  differential  calculus  the  rate  at  which  V  varies  as  x 
increases  equals 

-A/x*+B/(p—  x)*    ......     (7). 

This  vanishes  when  x=p/{  1+(J5/M)1/2}.  Since  x  must 
be  less  than  p  we  take  the  positive  sign,  and  it  is  easy  to 
see  that  when  x=p/{  l-\-(B/A)l/2},  V  attains  its  minimum 
value.  This  can  be  seen  from  first  principles  as  follows. 
When  x  is  very  small  the  amount  of  copper  used  in  Ma 
(Fig.  17)  must  be  excessive.  As  x  increases,  (7)  shows  that 
the  volume  of  copper  required  is  rapidly  diminishing.  It 
attains  its  minimum  value  when  (7)  vanishes,  and  when  x 
is  nearly  equal  to  p,  the  volume  required  is  again  very  large 
as  the  voltage  drops  in  ab  and  ac  have  to  be  very  small. 


DISTRIBUTING  NETWORKS  77 

The  rule  therefore  is  to  choose  the  cross  sections  so  that 

}V*]  .  .(8). 


(9) 


Hence,  by  (3)  we  must  make 

S  =  {dC+l(C'+C")}/30x,     \ 
S'=d'C'/W(p-x), 
and  S"=d"C"/W(p—  x),  ) 

where  the  value  of  x  is  computed  from  (8). 
Numerical         -Let  the  numerical  data  of  the  problem  be 
example       ag  gjven  m  pjg     18?   so  t^at  we  have  ^  =  100, 

d'=300,   <T=80,  1=250,    Z/=400,   T=200,  (7=40,  C'  =  15, 


g« 


FIG.  18. 

(7" =20,    the   lengths    being   measured  in   metres,  and   the 
currents  in  amperes.     Let  us  also  suppose  that  p  is  4  volts, 
and  that  Ma  and  ab  are  to  have  the  same  section  S. 
We  have,  therefore,  by  (5) 

8  =  { 100  x  40+250  x  35+300  x  15}  /(30  x  4) 

=  143*75  sq.  mms. 

We  have  also,  by  (6),  if  p  is  to  be  the  voltage  drop  from 
M  to  c, 


=  143-75(80  x  20/300  x  15) 
=  51«1  sq.  mms. 
It  is   interesting   to  compare   these   numbers   with  the 


78 


ELECTRIC  CABLES  AND  NETWORKS 


numbers  obtained,  by  (8)  and  (9),  for  the  most  economical 
solution.     From  (8),  we  find  that  x  equals 
4/[l-H300.15.400+80.20.200)1/2/(100.40.250+2502.35)1/2], 
which  is  nearly  equal  to  2-2. 
Hence,  by  (9), 

£  =  {100.40+250.35}/66  =  193  approx. 
8'  =300. 15/54  =83-3 

£"=80.20/54  =29-6       „ 

The  first  solution  requires  2,074,000  cubic  cms.  of  copper 
and  the  second  1,750,000  cubic  cms.  A  saving  of  about 
16  per  cent,  in  the  quantity  of  the  copper  used  could  thus 
be  made  by  adopting  the  second  solution. 


M 


I' 


FIG.  19. 

Let  us  suppose  that  the  feeding  centres  M 

from  both     and  L  (Fig.  19)  are  at  the  same  potential,  and 

Single        that  the  resultant  current  C  branches    off    at 

a.     If  a;  be  the  current  entering  at  the  feeding 

centre  M9  and  C — x  be  the  current  between  L  and  a,  then 

since  the  voltage  drop  p  from  M  to  a  equals  the  voltage 

drop  from  L  to  a,  we  have 

(pl/S)x=(pl'/S)  (C-x)=p, 
and  therefore,  x=C 

We  also  have, 

S=xl/30p 


(10), 


=Cl(L—l)/3QpL 


DISTRIBUTING  NETWORKS 


79 


where  L=l-\-l'.     Hence  S  has  its  maximum  value  when 
l=L/2. 

If  l=L/2,  the  currents  in  I  and  V  are  each  equal  to  (7/2, 

and  by  (10),  S=CL/l20p.     If  Z=L/4,  x  is  3(7/4,  (7— »  is 

(7/4,  andfl=(12/16)OL/120p.     If  l=L/S,  x  is  7(7/8,  C—x 

is  (7/8,  and  #=(7/16)  CL/l20p.     By  supposing  £  and  M 

to  be  coincident,  we  see  that,  except  in  the  case  when  a 

is  midway  between  M  and  L,  more  copper  is  required  when 

the  distributing  centre  is  fed  from  two  centres  than  when 

it  is  fed  from  the  nearer  one  only.     We  have,  however, 

an  additional  security  for  the  continuity  of  the  supply. 

Two  centres        ^et  us  now  consider  the  case  of  two  feeding 

distributing     centres   M  and  L  (Fig.  20)  supplying  a  distri- 

CaDbranchd     buting  centre  at  a  and  a  branch  main  at  b. 

mam        Let  the  current  required  at  a  be  C2  and  that 


M 


C,-£C 


C.+X 


FIG.  20. 

at  b  be  (7^  Let  also  C2  —  x  be  the  current  in  Ma,  and  Ct+x 
be  the  current  in  Lb.  We  shall  suppose  that  the  section 
S  of  the  main  joining  M  and  L  is  uniform.  Then,  since 
the  voltage  drop  from  M  to  a  must  be  equal  to  that  from 
L  to  a,  we  have,  by  (4) 


and  thus,  x=  (1C,—  l"Ci)/(l+l'+l")    ..      ..     (11), 

where  l=Ma,  l'=ab  and  l"=bL. 

From  (11)  we  see  that  if  IC2  be  greater  than  T(7l3  x  is 


80          ELECTRIC  CABLES   AND   NETWORKS 

positive,  and  the  direction  of  the  current  is  as  indicated 
in  Fig.  20.     When,  however,  IC2  is  less  than  VC\,  the  current 
is  in  the  reverse  direction.     It  is  also  to  be  noticed  that  the 
value  of  x  is  independent  of  the  size  of  the  section  either  of 
the  main  or  the  branch  main. 

We  shall  now  find  the  areas  of  the  cross  sections  of  the 
mains  so  that  the  volume  of  the  copper  used  in  them  may 
be  a  minimum.     The  size  of  the  submain  is  determined 
when  the  voltage  drop  p±  at  b  is  known,  for  the  maximum 
voltage  drop  at  the  end  of  the  submain  must  not  exceed 
p.     Hence  if  S"  be  the  area  of   the  cross  section  of    the 
submain,  we  have  by  (3), 

S"=d"C,/W(p—p,)  ......     (12), 

where  d"  is  the  distance  of  the   distributing  centre   of  the 
load  Ci  from  b.     By  (3),  we  have 

S=r(Ci+x)/3QPi  ......     (13), 

and  hence  the  volume  of  the  copper  required,  namely, 


where  LI  =?+£'+£",  and  L2  is  the  length  of  the  submain, 
equals 


By  the  differential  calculus  this  has  a  maximum  or  a  mini- 
mum value  when 


and  it  is  easy  to  see  that  when 


the  volume  is  a  minimum.     We  can  readily  find  pt  by 
this  formula,   and  hence,   S  and  S"  are    determined    by 

(13)  and  (12). 

It  is  to  be  noticed,  however,  that  the  value  of  p^  found  by 

(14)  may  make  the  voltage  drop  at  a  greater  than  p    and 
this  is  not  permissible.     Since  the  voltage  drop  at  a  equals 
pi+pJ,'x/l"(Ci+x),  we  see  that  this  occurs  when 


DISTRIBUTING  NETWORKS 


81 


I'x/l'fa+x)     is     greater      than 
or      LJ'*x*  is  greater  than  LJ'd'C^d+x)  . .         (15). 

In  this  case  the  most  economical  solution  is  to  choose  S, 
so  that 

S=l(C2—x)/Wp          (16), 

and  also,  p,=pl/f(C1i-{-x)/l(C2—x)       ..      ..     (17). 

We  calculate  pt  by  (17),  and  then  S"  is  found  by  (12). 

Let  us  now  consider  how  to  calculate  the 

from  one      cross  section  of  a  loop  of  cable   (Fig.  21)  fed 

centre        from   the   centre   M.     Let   the   values   of    the 


currents  be  as  marked  in  the  diagram.  We  have  marked 
the  arrow-heads  as  if  the  current  were  flowing  in  the  same 
direction  all  round  the  loop.  This  is  merely  done  to  obtain 
algebraical  symmetry  in  our  equations.  The  value  of 
x — C2 — C3,  for  instance,  is  always  negative.  Since  the 
P.D.  between  a  and  c  added  to  the  P.D.s  between  c  and  6, 
and  between  b  and  a  must  equal  zero,  we  have 

(p/S)  { liX+l2(x—C2)  +13(X—C2—C3) }  =0, 
and  therefore, x  =  {12C2 -H3(C2+C3)} /(k  +12+13)  . .     (18), 
where  Il3 12 ,  and k ,  are  the  lengths  of  ac,  cb,  and  ba,  respectively. 

G 


82          ELECTETC  CABLES  AND  NETWORKS 

Let  us  suppose  that  this  value  of  x  is  less  than  C2.  In 
this  case  the  potential  will  have  its  minimum  value  at  C, 
and  the  potential  drop  between  M  and  C  will  be  p.  Let 
Pi  be  the  P.D.  between  M  and  a.  Then  if  S  be  the  section 
of  the  main  Ma  and  S'  be  the  section  of  the  cable  forming 
the  loop,  we  have,  by  (3), 


and  S'=Zi2?/30(p—  pt)  J 

where  I  is  the  length  of  the  main  M  a,  and  d  is  the  distance 
of  the  feeding  centre  for  Ci  from  M  .  Hence  if  F  be  the 
volume  of  the  copper  used  in  the  main  M  a  and  in  the  loop 
abc,  we  have 


_  m    ,       n 
-j-  —     — j 

^i       2^ — PI 
by  (18)  and  (19), 

where  m=l{d,C, +l(C2+C3,,,  .  -   , 

and  n  =Z,  { Z2Cr2+Z3(C'2  +^3) }  /30/ 

Now  m,  7i  and  p  are  independent  of  the  values  of  the  sections 

of  the  mains,   and  hence  by  the  differential  calculus.    F 

will  have  its  extreme  values  when 

m  n 


and  when  p^  =  p/  { 1  +  Jn/m }        (21), 

the  volume  of  the  copper  employed  in  the  mains  has  its 
minimum  value.  Having  found  the  value  of  p^  from  (21), 
the  values  of  S  and  S'  can  be  readily  found  from  (19). 

Loop  with  In   the   lo°P    (Fig-  22)   let  A  M>   °  and  N  be 

feeding       ^e  Ceding  centres  which  we  suppose  are  all 

centres       maintained  at  the  same  potential.     Let  x  be 

the  current  in  Ma,  and  let  currents  d,  C2  and  C3  be  tapped 

from  the  loop  at  points  a,  b,  and  c,  between  M  and  L.     Then, 

if  Ma=li,  ab=l2,  bc=l3  and  cL=li}  we  have 


DISTRIBUTING  NETWORKS  83 

)lz +(x—Cl—C2)h+(x—Ci—C.f—C3)lt  =0, 
and  therefore, 

X  =  {C1(12  +  13  +  1,)+C3(13  +  1,)+C31,}/(11+12  +  13^1.). 
If  the  value  of   x  found  from   this  equation  be  less   than 


M 


FIG.  22. 

<7i,  a  will  be  the  point  of  minimum  potential.  If  x  be 
greater  than  d,  but  less  than  Ci-\-C2,  &  will  be  the  point 
of  minimum  potential  and  if  x  be  greater  than  Ci+C2,  c 
will  be  the  point  of  lowest  potential  between  L  and  M  . 
Let  us  first  suppose  that  x  is  less  thand.  In  this  case,  by 
(3),  S=  liX/SOp.  If  the  value  of  x  lies  between  Ct  and 
Oi-\-C2)  the  section  of  the  loop  between  M  and  L  would 
be  given  by 


and  when  the  value  of  x  is  greater  than  (7i+(72,  the  equation 
for  S  is 

S  =Z4(d  +C2+Cs—x)/30p. 

Ring  main         We   shall   now   consider  the   case   of  a  ring 

feeding       main   and   in   order   to   simplify   the   formulae 

we  shall  suppose  that  it  forms  a  circle  (Fig.  23), 

with  the  power  station  S  at  its  centre,  and  that  the  feed- 


84          ELECTRIC  CABLES   AND  NETWORKS 

ing  centres  are  equally  spaced  round  it.  We  shall  also 
suppose  that  the  load  is  evenly  distributed  so  that  the 
points  of  minimum  potential  are  midway  between  the 
feeding  centres.  If  there  are  n  feeders,  and  C  is  the  total 


A, 


FIG.  23. 

current  output,  C/n  will  be  the  current  in  each  feeder  and 
half  (C/2n)  of  this  current  will  flow  in  one  direction  round 
the  circle  and  half  in  the  other. 

Let  pi  be  the  drop  of  potential  from  8  to  any  of  the 
feeding  points.  Then,  by  (3),  the  section  of  each  feeder 
is  given  in  square  millimetres  by 


where  a  is  the  radius  of  the  circle  in  metres. 

The  section  S'  of  the  ring  main,  in  square  millimetres,  is 
given  by  (see  p.  72) 

S'=(C/2n)  (27ra/2n)/QO(p—pi). 

Hence,  if  V  be  the  volume  of  the  copper  required  in  cubic 
centimetres,  we  have 
V/2  =n(C/n)a^/30 

=Ca2/30pi  -f  <7a 
By  the  differential  calculus,  V  has  its  minimum  value  when 
pi=np  v/2/(7r  -\-n  v/2). 


DISTRIBUTING  NETWORKS 


85 


In  this  case, 


If  n  were   infinite,  the  volume  V  of  the  copper  required 
would  equal  2<7a2/30p,  and  thus 


The  following  table  shows  how  this  ratio  varies  as  n  increases. 


n 

1 

2 

3 

4 

5 

6 

1 

8 

9 

10 

100 

v/v 

10-4 

4-46 

3-03 

2-42 

2-09 

1-88 

1-73 

1-63 

1-56 

1-49 

1-04 

The  proper 

the^power      anc^ 
station  of 


It  will  be  seen  that  a  substantial  saving  in  copper  is 
effected  by  increasing  the  number  of  the  feeders. 

When  the  positions  of  the  feeding  centres 
curren^s  they  require  are  fixed,  the 
feeders  varies  largely  with  the  site 
of  the  generating  station.  We  shall  now  prove  that  the 
most  economical  site  is  the  "  centre  of  gravity  "  of  the 
various  loads  at  the  various  feeding  centres.  By  the 
centre  of  gravity  of  the  load  is  meant  the  centroid  of 
masses,  proportional  to  the  loads  at  the  various  feeding 
centres,  placed  at  these  centres. 

Let  us  suppose  that  Ai}  A2,  .  .  .  An  (Fig.  24)  are  the 
feeding  centres,  and  that  d,  G2,  .  .  .  Cn  are  the  currents 
required  for  them.  Then,  if  p  be  the  maximum  permissible 
voltage  drop  in  the  mains  between  the  generating  station 
S  and  the  feeding  centres,  the  section  Si  of  the  main  joining 
S  and  A!  is  given  by 


and  the  volume  of  this  main  by  2(7iZi2/30p.     Hence,  if  V 
be  the  total  volume  of  the  copper  required 


86         ELECTRIC  CABLES  AND  NETWORKS 

Now  it  is  a  well-known  theorem  in  statics  (see  Thomson  and 
Tait's  Elements  of  Nat.  Phil.  §  196)  that  SCI2  has  a  mini- 
mum value  when  S  coincides  with  the  centroid  G  of 
masses  Ci,  C2,  .  .  .  Cn  placed  at  AI,  A2,  .  .  .  An  re- 


spectively.  If  Vm  denote  the  minimum  value  of  F,  it  also 
readily  follows  that 

(F— F.)/2=(01+0,+.    .    .+CJQ8*/30p, 
and  this  is  the  volume  of  the  copper  saved  by  moving  the 
generating  station  from  S  to  G. 

It  is  to  be  noticed  that  we  have  chosen  the  volume  of 
the  copper  in  the  two  cases  so  that  the  power  expended 
in  the  mains  namely  p2C  is  the  same  in  the  two  cases. 

Let    us    suppose    that    the    feeding    centres 
Example 

A i,  A 2,   ..    .   An  were  equally  spaced   round 

a  circle  of  radius  a,  and  that  a  current  C  was  required  at 
each.  Then,  if  the  generating  station  were  at  the  centre 
of  this  circle,  the  volume  of  the  copper  required  would  be 
given  by 


DISTRIBUTING  NETWORKS  87 

If  S  were  at  a  distance  lea  from  G,  the  volume  F  of  the 
copper  required  would  be  found  from 


It  is  therefore  very  important  in  practice  that  Ic  should 
be  small. 

The  feeding        ^et  ML  (Fig.  25)  be  a  straight  main  which 

Straight     we  suppose  to  be  uniformly  loaded  and  let  A 

be  the  position  of  the  generating  station.     It 

is  required  to  find  the  position  of  a  point  F  in  M  L,  so  that 

A 


L-x. 
FIG.  25. 


when  ML  is  of  uniform  section,  the  copper  required  for 
the  feeder  AF  and  the  main  ML  may  be  a  minimum, 
subject  to  the  condition  that  the  voltage  drop  from  A  to 
the  farthest  point  of  ML  must  not  be  less  than  p.  From 
A  (Fig.  25)  draw  AN  at  right  angles  to  LM  or  LM  pro- 
duced. If  we  take  any  point  F  at  a  distance  x  from  M 
as  the  feeding  point  and  if  we  suppose  that  x  is  less  than 
1/2,  L  will  be  the  point  of  minimum  potential.  We  shall 
now  find  the  sections  of  the  feeder  AF  and  the  main  ML 
so  that  the  copper  used  in  them  is  a  minimum,  the  voltage 
drop  from  A  to  L  being  p.  Let  $1  be  the  section  of  the 
feeder  AF  and  y  its  length.  Then,  by  (3), 


where  pi  is  the  voltage  drop  between  A  and  F.     If  we 


88         ELECTRIC  CABLES  AND  NETWORKS 

suppose  the  main  ML  to  be  uniformly  loaded,  the  section 
$2  will  be  given  by 


where  I  is  the  length  of  ML.     Hence  the  volume  F  of  the 
copper  in  cubic  centimetres  is  given  by 

V/2=Cy*/30Pl+C(l—  xy/W(p—  p,), 

The  volume  of  the  copper  required,  therefore,  varies  as 
PI,  and  has  its  extreme  values  when 


and  hence,  \/2y/pi  =  (l  —  x)/(p  —  p 


the  positive  sign  being  taken  as  this  gives  the  only  ad- 
missible value,  and  in  this  case  V  has  its  minimum  value  F'. 
Hence  the  minimum  possible  volume  V  of  copper  when 
the  feeding  centre  is  F,  is  given  by 


—  x+  {  2d 

We  have  now  to  find  out  what  position  of  F  makes  this 
the  absolute  minimum. 

By  the  differential  calculus  it  follows,  almost  at  once, 
that  when  x  equals  d  —  a,  V  is  the  absolute  minimum 
F  and  hence 


miTC  , 


It  is  to  be  noticed  that  in  Fig.  27  we  have  taken  a  positive 
when  N  is  to  the  left  of  M  .  If,  therefore,  N  lies  between 
M  and  C  (Fig.  25)  at  a  distance  a  from  M,  MF  —d-\-a  when 
the  volume  has  its  minimum  value  (C/3Qp)  (I  —  a-\-d)z. 
Now  MF  cannot  be  greater  than  1/2  or  our  assumption 
that  the  minimum  potential  is  at  L  is  no  longer  true.  We 
see,  therefore,  that  if  d  —  a,  when  N  is  to  the  left  of  M,  or 
d-\-a,  when  N  lies  between  M  and  C,  be  not  greater  than 
1/2  the  most  economical  solution  is  to  make  x  equal  to 
d  —  a  or  d-\-a  according  as  N  is  to  the  left  or  right  of  M. 


DISTRIBUTING  NETWO&KS  89 

If  the  given  quantities  be  greater  than  1/2,  then,  the  most 
economical  solution  is  to  make  the  middle  point  C  of  ML 
the  feeding  centre.  Lastly  when  N  is  to  the  left  of  M 
and  d — a  is  negative,  M  is  the  proper  feeding  centre. 
Practical  From  the  symmetry  of  the  arrangement 
rule  when  M  lies  to  the  right  of  C  similar  solutions 
apply  in  the  various  cases.  The  analytical  results  lead 
to  the  following  practical  rule  for  finding  the  feeding  centre 
for  a  straight  main  ML,  when  the  distributing  centre  is  at 
any  point  A  (Fig.  25).  Draw  AN  at  right  angles  to  LM  or  LM 
produced.  Make  the  angle  NAF  equal  to  45°,  where  F 
lies  on  LM  or  LM  produced.  Then  if  F  lie  between  N 
and  M,  M  is  the  feeding  centre,  but  if  it  lie  between  M 
and  C,  or  on  C,  F  is  the  feeding  centre.  Finally,  if  it  lie 
to  the  right  of  C,  C  is  the  feeding  centre.  The  following 
is  a  graphical  illustration  of  the  rule. 


Let  ML  (Fig.  26)  be  the  main  which  we  suppose  to  be 
uniformly  loaded.  Make  the  angle  M'Ma  equal  to  45° 
and  draw  Cb  parallel  to  Ma,  where  C  is  the  middle  point 
of  ML.  Similarly  make  the  angle  L'La'  equal  to  45° 
and  draw  Cb'  parallel  to  La'.  Let  us  suppose  that  the 
generating  station  A  is  above  the  line  ML.  If  A  lie  within 
the  angle  M'Ma,  M  is  the  feeding  centre.  If  it  lie  between 
the  parallel  lines  Ma  and  Cb,  then  F  is  the  feeding  centre, 


90          ELECTETC  CABLES   AND  NETWORKS 

where  AF  is  parallel  to  Ma.  If  it  lie  within  the  right 
angle  bCb',  C  is  the  feeding  centre.  If  it  lie  between  Cb' 
and  La'  we  draw  AF  parallel  to  Cb',  and  finally  if  it  lie 
within  the  angle  a'LL  ',  L  is  the  feeding  centre.  An  ex- 
actly similar  solution  applies  when  A  is  below  the  line  ML. 
When  the  foot  of  the  perpendicular  from  A  on  ML  falls 
between  M  and  C'  where  C"  is  the  middle  point  of  M  C,  we 
have 


Now  at  all  points  on  M  T,  d  equals  —  a,  and  thus  if  the 
generating  station  A  be  situated  on  MT,  Vmint  equals 
(C/3Qp)l2  and  is  therefore  constant.  If  with  centre  C 
and  radius  CT  we  describe  the  quadrant  TT'  of  a  circle, 
then,  if  A  be  situated  at  any  point  on  this  quadrant,  Fmiw> 
will  have  the  same  value.  Consequently,  if  A  be  situated 
at  any  point  inside  MTT'LCM,  Vmin,  will  be  less  than  if 
it  were  situated  at  M9  and  if  A  be  situated  above  MTT'L, 
Vmin.  wul  be  greater  than  if  A  were  at  M  . 

It  is  now  easy  to  see  that  the  locus  of  A  for  which  Fmin> 
is  constant  is  a  quadrant  of  a  circle  between  Cb  and  Cb',  a 
straight  line  parallel  to  M  T  between  Cb  and  Ma,  a  quad- 
rant of  a  circle  between  Ma  and  TM  produced,  etc. 

In  practice,  in  order  to  reduce  the  initial 
cost  of  the  copper  required  when  designing 
a  distributing  network,  it  is  customary  in  certain  cases 
to  put  a  "  boosting  "  dynamo  or  "  booster  "  in  series  with 
a  feeder,  so  as  to  maintain  the  potential  of  the  distributing 
centre  constant  however  the  load  may  vary.  A  booster 
(Fig.  27)  has  two  directly  coupled  rotating  armatures. 
One  of  these  is  the  armature  of  a  shunt  wound  motor 
driven  from  the  mains,  the  other  the  armature  of  a  series 
dynamo  connected  in  series  with  the  feeder. 

When  no   current  is  passing  through  the  dynamo,  the 


DISTRIBUTING  NETWORKS  91 

field  is  practically  unexcited  and  the  E.M.F.  generated 
by  the  rotating  armature  is  negligibly  small.  When, 
however,  there  is  a  current  in  the  feeder  the  field  magnets 


FIG.  27. — Direct  current  booster. 

are  excited  and  an  E.M.F.  e  is  generated.  If  R  be  the 
resistance  of  the  dynamo  windings  and  of  the  outgoing  and 
return  feeder,  C  the  current,  and  E  the  initial  potential  of 
the  feeding  point,  then  the  new  potential  will  be  E-\-e — CE. 
If  the  first  part  of  the  characteristic  of  the  dynamo 
be  a  straight  line,  it  is  possible  to  arrange  that  e — CR  is 
practically  zero  for  all  the  values  of  the  current  during 
normal  working. 

The  power  expended  in  the  feeding  circuit  is  C2R  and 
we  have  now  to  consider  whether  it  is  more  economical 
to  use  a  booster  or  to  increase  the  weight  of  the  feeder. 

Let  us  suppose  that  the  booster  is  so  designed 

The 

economy  of    that  at  full  load  the  drop  p  of  the  potential 
at  the  distributing  centre  is  the  same  as  if  a 
single  feeder  of  resistance  R  were  used.     Let  us  suppose 
also  that  e=np.     Then,  at  full  load,  we  have 

np — CBi  = — p. 

or  CE±=(n+l)p9 

and  thus,  Ei=(n+l)E. 

Hence,  when  a  booster  is  used,  the  copper  required  is  only 
the  (n-\-l)ih  part  of  that  required  for  a  feeder  main  by 
itself.  It  has  to  be  remembered  that  the  losses  will  be 
(n-j-1)  times  greater,  but  they  are  only  heavy  at  full  load. 
Hence,  for  a  small  distributing  centre  at  a  considerable 


92          ELECTRIC  CABLES  AND  NETWORKS 

distance  from  the  station,  the  use  of  a  booster  often  effects 
considerable  economies.  If  the  interest  saved  on  the 
initial  cost,  by  using  the  booster  and  the  lighter  main,  be 
greater  than  the  annual  increase  of  the  generating  charges 
together  with  the  cost  of  the  maintenance  of  the  booster, 
it  will  be  more  economical  to  use  a  booster.  When  the 
distributing  centre  is  large  a  special  dynamo  must  be  used. 

REFERENCES. 

J.  Herzog  and  L.  Stark,  "  Ueber  die  Stromvertheilung  in  Leitungs- 
netzen."  Elektrotechnische  Zeitschrift,  vol.  ii.  pp.  221  and  445, 
1890. 

O.  Li  Gotti,  "  Sur  une  Methode  pour  le  Calcul  des  Reseaux  de  Dis- 
tribution." Eclairage  Blectrique,  vol.  44,  pp.  281-286,  1905. 

Marcel  Leboucq.  "  Methode  Pratique  de  Calcul  de  Reseau  Blec- 
trique,  d'Eclairage  et  de  Transport  de  Force."  Societe  Beige 
d'Electriciens,  Bulletin,  23,  pp.  109-137,  1906. 


INSULATION    RESISTANCE    OF    HOUSE 

WIRING 


CHAPTER    V 

Insulation    Resistance  of  House   Wiring 

Institution    Rules — Ohmmeter    and    Generator — Megger — Electro- 
static voltmeter  method — Earth  lamps — References. 

WHEN  a  building  has  been  wired  for  the  electric  light  it 
is  necessary  to  make  certain  electrical  tests  to  find  out 
whether  the  mains  are  properly  insulated  from  one  another 
and  from  earth.  In  the  wiring  rules  (1907)  issued  by  the 
Institution  of  Electrical  Engineers  the  methods  of  testing, 
etc.,  are  described  as  follows — 

Institution  "97.  The  insulation  resistance  to  earth  of 
rules  the  whole  or  any  part  of  the  wiring  must,  when 
tested  previously  to  the  erection  of  fittings  and  electroliers, 
be  measured  with  a  pressure  not  less  than  twice  the  intended 
working  pressure,  and  must  not  be  less  in  megohms  than 
30  divided  by  the  number  of  points  under  test.  For  this 
purpose  the  points  are  to  be  counted  as  the  number  of 
pairs  of  terminal  wires  from  which  it  is  proposed  to  take 
the  current,  either  directly,  or  by  flexibles,  to  lamps  or 
other  appliances. 

"98.  Current  must  not  be  switched  on  until  the  follow- 
ing test  has  been  applied  to  the  finished  work  : — 

"  The  whole  of  the  lamps  having  been  connected  to  the 
conductors  and  all  switches  and  fuses  being  on,  a  pressure 
equal  to  twice  the  working  pressure  must  be  applied  and 


96          ELECTRIC   CABLES   AND  NETWORKS 

the  insulation  resistance  of  the  whole  or  any  part  of  the 
installation  must  not  be  less  in  megohms  than  25  divided 
by  the  number  of  lamps.  When  all  lamps  and  appliances 
have  been  removed  from  the  circuit,  the  insulation  resist- 
ance between  conductors  must  not  be  less  than  25  megohms 
divided  by  the  number  of  lamps.  The  insulation  of  any 
individual  sub-circuit  must  not  fall  below  1  megohm. 
Any  motor,  heater,  arc  lamp  or  other  appliance  may  be 
connected  to  the  supply  of  electrical  energy  provided  that 
the  insulation  of  the  parts  carrying  the  current  measured 
as  above,  is  greater  than  1  megohm  from  the  frame  or 
case. 

"  99.  The  value  of  systematically  inspecting  and  test- 
ing apparatus  and  circuits  cannot  be  too  strongly  urged. 
Records  should  be  kept  of  all  tests,  so  that  any  gradual 
deterioration  of  the  system  may  be  detected.  Cleanliness 
of  all  parts  of  the  apparatus  and  fittings  is  essential. 

"  100.  Before  making  any  repairs  or  alterations,  the 
circuits  which  are  being  attended  to  must  be  entirely  dis- 
connected from  the  supply." 

It  is  advisable  to  make  two  insulation  tests  between 
the  mains.  For  the  first  test  all  the  switches  should  be 
turned  off  and  all  lamps  and  appliances  should  be  in 
position.  The  result  of  this  test  will  show  whether  any 
switch  is  faulty  or  not.  A  second  test  should  be  made 
with  the  switches  turned  on  and  all  the  lamps  and  appli- 
ances removed.  This  will  show  whether  the  insulation 
resistance  between  the  "  flexibles  "  connecting  the  ceiling 
roses  with  the  lamp  holders,  etc.,  is  satisfactory. 

The  results  of  insulation  tests  give  only  a  partial  indi- 
cation of  the  way  in  which  the  wiring  of  a  building  has 
been  done  and  the  quality  of  the  materials  used.  If 
the  house  be  damp  the  insulation  will  probably  come  out 


INSULATION  RESISTANCE  OF  HOUSE  WIRING  97 

low  no  matter  how  carefully  the  wiring  has  been  done. 
If  the  house  be  dry  the  insulation  resistances  will  probably 
come  out  very  high  even  although  the  insulating  materials 
used  be  of  poor  quality  and  the  joints  be  made  in  the 
most  careless  manner. 

The  forty-first  of  the  Board  of  Trade  Regulations  for 
securing  the  public  from  a  "  bad  and  inefficient  supply  of 
the  electric  light  "  is  as  follows  : — 

"  The  undertakers  shall  not  connect  the  wiring  and 
fittings  on  a  consumer's  premises  with  their  mains  unless 
they  are  reasonably  satisfied  that  the  connexion  would  not 
cause  a  leakage  from  those  wires  and  fittings  exceeding 
one  ten  thousandth  part  of  the  maximum  supply  current 
to  the  premises  ;  and  where  the  undertakers  decline  to 
make  such  connexion  they  shall  serve  upon  the  consumer 
a  notice  stating  their  reason  for  so  declining." 

This  is  usually  taken  to  mean  that,  if  V  be  the  declared 
pressure  at  the  consumer's  terminals  and  F  be  the  insulation 
resistance  to  earth  of  the  house  wiring,  V/F  must  be  less 
than  the  ten  thousandth  part  of  the  maximum  supply 
current.  V/F,  however,  is  a  purely  imaginary  current. 
To  make  this  clear  we  shall  consider  the  case  of  a  house 
the  wiring  of  which  is  connected  with  two  of  the  mains 
of  a  direct  current  3- wire  system  of  supply. 

On  open  circuit,  the  potentials  to  earth  of  the  house 
mains  are  the  same  as  the  potentials  of  the  supply  mains 
to  which  they  are  attached.  As  the  potential  difference 
drop  enclosed  circuit  is  at  the  most  2  per  cent.,  we  see  that 
no  great  error  is  made  by  the  assumption  that  the  potential 
to  earth  of  a  main  is  constant  at  all  points  of  its  length 
whatever  may  be  the  load.  In  order  to  simplify  the  theory 
we  shall  make  this  assumption.  As  the  insulation  resist- 
ance of  the  coverings  of  the  mains  is  not  infinite,  leakage 

H 


98 


ELECTRIC  CABLES  AND  NETWORKS 


currents  will  always  be  flowing  either  from  the  copper  to 
the  earth  or  vice  versa.  It  is  convenient  to  divide  the 
paths  of  the  leakage  current  into  three  groups.  In  a  path 
of  the  first  group,  the  current  flows  between  the  copper  of 
one  main  and  the  earth.  In  a  path  of  the  second  group 
it  flows  between  the  copper  of  the  other  main  and  the 
earth,  and  in  a  path  of  the  third  group  the  current  flows 
from  one  main  to  the  other  without  passing  through  the 
"  earth."  The  path,  for  example,  may  be  from  one  main 
to  the  other  across  the  surface  of  a  porcelain  switch  which 

may  be  excellently  in- 
P  N     sulated  from  the  earth. 

AA/VVVV P     The  point  of  this  path, 

therefore,  which  is  at 
zero  potential  must 
not  be  considered 
as  belonging  to  the 
"earth." 

Let  P  and  N  (Fig.  28) 
denote  the  cross  sec- 
tions of  the  conductors 
of  the  house  mains.  Let 
x  denote  the  resultant 

resistance  of  the  first  group  of  leakage  paths  which  we 
suppose  connects  P  with  the  earth  E.  Similarly  let  y 
denote  the  resistance  of  the  second  group  connecting  N 
with  E,  and  a  the  resistance  of  the  third  group  of  leakage 
paths  which  are  all  insulated  from  the  earth.  Strictly 
speaking  the  values  of  x,  y,  and  a  vary  with  the  number 
of  switches  closed  and  with  the  number  of  lamps  which 
are  taken  from  their  sockets.  To  fix  our  ideas  we  shall 
suppose  that  the  readings  are  taken  when  all  the  switches 
are  on  and  all  the  lamps  are  removed  from  their  sockets, 


E 

FIG.  28. 


INSULATION  RESISTANCE  OF  HOUSE  WIRING     99 

the  mains  being  put  in  metallic  connexion  during  the 
insulation  test  to  earth.  In  this  case  the  insulation  re- 
sistance F  to  earth  is  given  by 

F=xy/(x+y), 
and  the  insulation  resistance  R  between  the  mains  by 


The  values  of  x,  y,  and  a,  therefore,  cannot  be  determined 
from  a  knowledge  of  F  and  R  only. 

If  Fi  and  F2  be  the  potentials  of  the  mains  P  and  N, 
the  leakage  current  from  the  main  P  to  earth  will  be  V±/x 
and  the  leakage  current  from  N  to  earth  will  be  Vs/y.  We 
shall  also  have  a  leakage  current  (Fi  —  F2)/a  beween  the 
mains.  It  is  not  clear,  however,  whether  the  greatest  of 
these  currents  taken  singly  or  the  sum  of  the  numerical 
values  of  the  three  is  the  "  leakage  current  "  specified  by 
the  Board  of  Trade  rules. 

From  the  point  of  view  of  the  public,  the  considerations 
which  limit  the  magnitudes  of  the  leakage  currents  are  the 
risk  of  fire  and  the  damage  done  by  electrolysis.  The 
fire  risk  is  the  more  important.  From  this  point  of  view 
the  heating  effects  which  are  measured  by  Vf/x,  V22/y 
and  (Fi  —  F2)2/a  respectively  govern  the  danger.  If  the 
rules  are  to  be  equitable,  the  maximum  permissible  heating 
effects  should  be  the  same  in  all  cases.  If  we  double  the 
voltage,  therefore,  the  insulation  resistance  should  be 
quadrupled.  It  is  to  be  noticed  that  for  given  values  of 
x,  y  and  a  the  danger  will  be  less  the  more  distributed  are 
the  leakage  paths,  and  the  danger  will  be  greatest  when 
the  leakage  paths  are  concentrated  at  one  spot.  If,  how- 
ever, the  values  of  x,  y,  and  a  are  sufficiently  high,  the  leakage 
power  will  be  so  small  that  there  is  no  danger  of  fire  even 
if  there  is  only  one  fault.  It  is  important,  therefore,  to 
know  their  values. 


100        ELECTRIC  CABLES  AND  NETWORKS 

Measure-          ^he  best  instrument  to  use  for  the  measure- 

Se^auft      ment  of  the  insulation  resistance  of  the  wiring 
resistances 


Q£  a  biding  js  a  portable  high  voltage  gener- 
ator and  an  ohmmeter.  These  are  combined  in  an  instru- 
ment called  the  "  megger  "  described  below.  The  method 
of  procedure  is  as  follows  :  — 

1.  Measure  the  resistance  X  between  P  and  E,  when 
N  is  connected  with  E  by  a  piece  of  wire.     A  water-pipe 
makes   an   excellent   earth   connexion.     In   practice   it   is 
customary  to  make  this  measurement  at  the  main  fuse 
block.     We  take  out  the  fuses  and  connect  one  terminal 
of  the  ohmmeter  to  the  end  of  the  house  main  P  where  it 
joins   the   fuse   block.     N  is   connected   with   the   water- 
pipe  and  so  also  is  the  other  terminal  of  the  ohmmeter. 
On  turning  the  handle  of  the  generator,  the  pointer  of  the 
ohmmeter  gives  the  value  of  X  directly. 

2.  Measure  the  resistance   Y  of  N  to  earth  when  P  is 
earthed. 

3.  Measure  the  insulation  resistance  F  between  P  and 
N  in  parallel  and  the  earth. 

Our  equations  are, 

......  (1), 

......  (2), 

and  l/x+l/y  =  l/F,  ......  (3). 

Hence,  by  addition,  we  find  that 


and  therefore,  by  (2),  we  have 


by  (1),  l/y=(—l/X+l/Y+I/F)/2, 

and  by  (3),  I/a=(l/X+l/Y—l/F)/2. 

The  reciprocals  of  x,  y,  and  a  are  thus  found  in  terms  of 

measured  quantities  and  so  x,  y,  and  a  can  be  found.     It 


INSULATION*  RESISTANCE  OF  HOUSE  WIRING   10 1 

is  to  be  noticed  that  when  the  sum  of  the  reciprocals  of 
two  of  the  quantities  X,  Y,  and  F  is  nearly  equal  to  the 
reciprocal  of  the  third  quantity,  a  small  percentage  error 
in  the  determination  of  any  of  them  will  make  a  large 
percentage  error  in  the  computed  value  of  one  of  the 
quantities  x,  y,  or  a. 

As  an  example  let  us  suppose  that  X ,  Y,  and  F  are  found 
to  be  1»98,  2-38  and  4'09  megohms  respectively.  In  this 
case  l/a?=(l/l'98—  l/2-38+l/4-09)/2 

=(0-5051— 0-4202+0-2445)/2 
=0-1647, 

and  therefore,  #=6O7  megohms.  The  above  calculation 
is  best  made  with  the  help  of  a  table  of  the  reciprocals  of 
numbers.  Similarly  we  find  that  y  =  12-5,  and  a  =2-94 
megohms.  The  fault  resistance,  therefore,  of  the  main 
N  is  practically  double  that  of  the  main  P.  Hence,  unless 
there  is  any  special  reason  to  the  contrary,  it  would  be 
advisable  to  connect  N  with  the  supply  main  which  is  at 
the  higher  potential. 

In  practice,  the  resistance  to  earth  xy/(x+y),  which 
equals  4O9  megohms,  and  the  insulation  resistance 
a(x-\-y)/(x -\-y-\-a),  which  equals  2*54  megohms,  are  the 
quantities  which  are  measured.  But  as  a  knowledge  of 
these  two  quantities  only  is  not  sufficient  to  enable  us  to 
find  out  the  values  of  x,  y,  and  a,  we  cannot  determine  the 
leakage  power  or  the  leakage  currents.  We  know  that  both 
x  and  y  are  separately  greater  than  the  insulation  resistance 
to  earth,  and  that  a  is  greater  than  the  insulation  resistance 
between  the  mains.  Hence  we  see  that  4-09  is  the  minimum 
possible  value  of  either  x  or  y,  and  that  a  is  not  less  than 
2-54.  Since,  however,  the  actual  values  of  x,  y,  and  a  can 
be  found  by  the  above  method  in  a  few  minutes  it  is  always 
advisable  to  find  them  as  they  give  important  information 


102        ELECTK1C  CABLES   AND  NETWORKS 

about  the  relative  values  of  the  insulation  resistance  of  the 
two  mains. 

In  making  the  above  test  we  have  supposed  that  the 
readings  are  taken  when  all  the  switches  are  closed,  and 
all  the  lamps  and  other  appliances  are  removed  from  their 
sockets  or  disconnected.  In  this  case  the  main  part  of  the 
leakage  is  generally  taking  place  across  the  flexible  wires 
used  in  the  fittings,  and  the  value  of  a  found  by  the  test 
corresponds  to  the  value  of  a  when  all  the  lights,  etc.,  in 
the  building  are  switched  on.  If  now  all  the  lamp  switches 
are  turned  off,  and  all  the  lamps  are  in  position,  a  new  test 
can  be  made  to  see  if  there  is  any  important  alteration 
in  the  values  of  x,  y,  or  a.  The  values  found  in  this  case 
enable  us  to  find  the  leakage  currents  when  all  the  switches 
of  the  consuming  devices  are  turned  off. 

There  is  still  a  possible  source  of  leakage  that  we  have 
not  yet  considered,  namely,  the  direct  leakage  between 
the  terminals  of  the  glow  lamp  itself.  The  terminals 
usually  consist  of  pieces  of  brass  separated  from  one  another 
and  from  the  collar  of  the  lamp  by  plaster  of  Paris.  If 
they  are  not  well  made,  there  may  be  considerable  leakage 
taking  place  between  the  terminals  or,  if  the  socket  for  the 
lamp  be  in  connexion  with  the  earth,  between  the  ter- 
minals and  the  collar.  Leakage  to  earth  through  the 
collar  of  a  lamp  lowers  the  apparent  fault  resistance  of  a 
main.  If,  however,  we  make  a  test  with  the  lamps  in 
position,  and  another  with  the  lamps  removed,  we  can 
easily  find  out  if  the  lamps  are  at  fault  between  the  leading 
in  wires  and  the  collar.  To  measure  the  insulation  re- 
sistance of  the  plaster  between  the  contact  pieces  is  diffi- 
cult as  they  are  directly  connected  by  the  filament.  It 
is  advisable,  therefore,  to  break  the  filament  of  a  sample 
lamp  in  order  to  test  this  resistance.  In  good  lamps  it 


INSULATION  RESISTANCE  OF  HOUSE  WIRING    103 

ought  to  be  exceedingly  high,  but  the  standard  of  1,000 
megohms  suggested  by  the  Engineering  Standards  Com- 
mittee (1907)  is  generally  considered  to  be  excessive. 


X 


CQ          O 


i- 


Ohmmeter 

and 
generator 


For  testing  the  insulation  resistance  of  the 
electric  wiring  in  a  building  an  ohmmeter  and 
generator  is  usually  employed.  The  generator 


104        ELECTRIC  CABLES   AND  NETWORKS 

consists  of  a  small  hand  dynamo  D  (Fig.  29)  enclosed  in  a 
portable  box.  Instruments  are  made  giving  pressures  of 
100,  200,  500,  or  1,000  volts.  Another  little  box  contains 
the  ohmmeter.  Two  coils  of  wire  A  and  B  (Fig.  29)  are 
placed  with  their  axes  making  a  fixed  angle  with  one 
another,  and  a  small  soft  iron  needle  ns  is  placed  between 
the  two.  The  connexions  for  testing  the  insulation  re 
sistance  of  the  mains  to  earth  are  shown  in  the  figure. 
When  the  handle  of  the  generator  is  turned  a  current 
passes  through  the  coil  B  and  the  resistance  in  series  with 
it.  If  the  insulation  resistance  of  the  cables  to  earth  be 
infinite  no  current  will  pass  through  A .  The  needle,  there- 
fore, will  set  itself  in  the  direction  of  the  resultant  mag- 
netic force  which  will  be  parallel  to  the  axis  of  the  coil  B. 
In  this  position  the  pointer  will  be  opposite  infinity  on  the 
scale  of  the  instrument.  Similarly  when  the  insulation 
resistance  to  earth  is  zero,  practically  all  the  current  will  pass 
through  A,  and  the  needle  will  be  parallel  to  the  axis  of 
this  coil,  the  reading  now  being  zero.  For  other  values  of  the 
insulation  resistance,  an  appreciable  current  passes  through 
both  coils,  and  the  needle  takes  up  an  intermediate  position. 
The  instrument  may  be  calibrated  by  putting  known  high 
resistances  between  its  terminals  and  turning  the  handle 
of  the  generator.  By  means  of  a  two-way  switch,  the 
resistance  in  series  with  B  can  be  altered  so  as  to  increase 
the  range  of  the  instrument.  In  practical  work,  the  read- 
ings can  be  trusted  to  within  2  or  3  per  cent. 

In  another  instrument  made  by  Messrs. 
Evershed  and  Vignoles,  called  the  megger, 
the  ohmmeter  and  generator  are  combined  so  that  they 
form  a  single  instrument.  The  manner  in  which  the 
ohmmeter  principle  is  applied  in  this  case  is  shown  in  Fig. 
30.  The  ohmmeter  and  the  generator  have  the  same 


INSULATION  RESISTANCE  OP  HOUSE  WIRING    105 

magnetic  circuit.  The  ohmmeter  has  two  coils  called 
the  pressure  and  current  coils.  They  are  mounted  on  a 
moving  axle  with  their  axes  inclined  to  one  another. 
The  field  in  the  annular  gap  in  which  the  current  coil 


FIG.  30.— The  Evershed  Megger. 

moves  is  uniform,  but  the  pressure  coil,  starting  from 
a  position  midway  between  the  poles,  is  dragged  into 
a  field  of  gradually  increasing  strength.  When  there 
is  no  current  in  the  current  coil,  the  pressure  coil 
is  at  rest  with  its  plane  midway  between  the  poles,  and 
the  pointer  reads  infinity.  If  the  resistance  be  zero,  a 
large  current  will  pass  through  the  current  coil,  and 
the  moving  system  will  be  dragged  round  by  the  forces 
acting  on  this  coil  into  a  new  position  of  equilibrium  where 
the  pointer  will  read  zero.  For  other  values  of  the  resist- 
ance and  therefore  of  the  current  in  the  current  coil,  the 
position  of  equilibrium  will  be  intermediate  between  these 
two  positions  and  the  pointer  will  give  definite  readings,  and 
so  the  scale  can  be  graduated.  By  suitably  designing  the 
shape  of  the  poles  so  that  the  resistance  offered  by  the 
magnetic  forces  acting  on  the  pressure  coil  to  the  motion 
increases  at  a  certain  rate,  instruments  with  open  and 
evenly  divided  scales  can  be  produced.  The  generators 


106         ELECTRIC  CABLES  AND  NETWORKS 

are  usually  wound  for  voltages  of  100,  200,  500,  or  1,000. 
The  low  range  instruments  read  from  0  to  100  megohms, 
and  the  high  range  instruments  from  10  to  2,000  megohms. 
In  order  to  eliminate  possible  errors  due  to  external 
fields,  a  differential  system  of  winding  is  adopted  for  the 
pressure  coil.  The  only  thing  that  has  to  be  guarded 
against  is  the  demagnetisation  of  the  magnetic  circuit. 

A  centrifugal  friction  clutch  is  sometimes  used  with  the 
generator  so  that,  when  it  runs  above  the  slipping  speed, 
its  velocity,  and  consequently,  the  E.M.F.  generated  is  very 
approximately  constant.  When  the  capacity  between 
the  circuits,  the  insulation  resistance  of  which  is  being 
measured,  is  greater  than  one  microfarad,  an  appreciable 
condenser  current  will  flow  through  the  current  coil  if  the 
E.M.F.  make  rapid  periodic  variations,  and  this  current 
will  affect  the  reading  of  the  instrument.  For  this  reason 
it  is  advisable  to  use  a  "  constant  pressure  megger  "  in 
these  cases.  Both  types  of  instrument  are  practically 
dead  beat.  As  the  total  weight  of  the  instrument  is  only 
about  18  Ibs.,  it  is  extremely  convenient  for  those  tests 
which  have  to  be  made  outside  the  testing  room. 
Electro-  Another  method  of  measuring  insulation 

voltmeter  resistance  is  by  means  of  an  electrostatic  volt- 
method  meter  and  a  known  resistance.  Let  us  suppose, 
for  example,  that  the  insulation  resistance  of  the  wiring 
of  a  building  has  to  be  measured,  and  let  the  two  mains 
and  earth  be  denoted  by  P,  N,  and  E,  respectively  (Fig.  28). 
The  procedure  is  as  follows  : — 1.  Disconnect  the  supply 
main  connected  with  N  from  the  fuse  box.  Let  the  volt- 
meter reading  between  P  and  N  and  between  N  and  E 
be  Fi  and  F2  respectively.  Then,  by  Ohm's  law,  we  have 

y/a  =  V2/V1        (1). 

2.  Disconnect   the  supply  main   connected  with  P  and 


INSULATION  RESISTANCE  OF  HOUSE  WIRING    107 

connect  the  other  main  again  to  N.  Let  the  voltmeter 
readings  between  N  and  P  and  between  P  and  E  be  F/ 
and  F/  respectively,  then 

3.  Finally,  without  altering  the  connexions  place  the 
resistance  r  between  P  and  N,  and  read  the  voltage  between 
the  same  points  again.  If  the  readings  be  now  F/'  and 
F2",  we  have 

x/{ar/(a+r)}=V2"/Vi"  ..      ..     (3). 

Hence,  from  (2)  and  (3), 

a=  r(Vi'/V*')  (VS/VS—  JY/JY). 

The  value  of  a  is  thus  found  and  the  values  of  x  and  y 
follow  readily  from  (1)  and  (2).  In  connexion  with  this 
method  three  small  carbon  resistances  1,  04,  and  0»01  of 
a  megohm,  will  be  found  useful. 

L__ , 

TT 


M 


X 

~  E 


FIG.  31. — Fault  indicator. 

The  following  method  of  automatically  indi- 

lamps"       eating  when  a  fault  occurs   on  either  of  the 
method 

mains    of     a    2-wire    distributing     system    is 

known  as  the  "  earth  lamps "  method.  Two  8-candle 
power  lamps  are  connected  in  series  between  the  mains 
at  the  distributing  board.  The  wire  joining  them  is  con- 
nected with  a  water  pipe  by  means  of  a  switch  82  (Fig.  31). 
If  this  switch  is  open  and  $1  is  closed  both  lamps  will 


108        ELECTRIC  CABLES  AND  NETWORKS 

burn  dimly  as  the  pressure  between  their  terminals  will 
only  be  half  that  of  the  supply  mains.  Let  us  now 
suppose  that  the  switch  S2  is  closed.  If  the  fault  resist- 
ance of  each  main  be  the  same,  no  change  in  the  relative 
brightness  of  the  lamps  will  ensue,  but  if  the  fault  resistance 
of  one  of  them  be  appreciably  lower  than  that  of  the  other, 
the  lamp  next  the  faulty  main  will  be  duller  than  the  other. 
On  a  100  volt  installation,  having  an  insulation  resistance 
greater  than  0-1  of  a  megohm,  the  effect  of  earthing  either 
of  the  mains  through  a  5,000  ohm  resistance  can  easily 
be  detected  by  the  earth  lamps. 

It  has  to  be  carefully  noticed,  however,  that  the  mere 
fact  that  opening  and  closing  the  switch  S2  has  no  appre- 
ciable effect  on  the  relative  brightness  of  the  lamps  is  not 
a  certain  indication  that  there  are  no  faults  on  the  mains. 
It  may  only  indicate  that  the  faults  are  equally  balanced 
between  the  two  mains.  If  the  lamp  connected  with  M 
glow  brightly  when  $1  is  open  and  S2  closed  this  will  show 
that  the  fault  resistance  of  L  is  small  compared  with  the 
resistance  of  the  lamp  (about  300  ohms). 

REFERENCES. 

A.  Russell,   "  Insulation  Resistance  and  Leakage  Currents."     The 

Electrician,  vol.   xli.,   p.   206,    1898. 
"  The  Megger,"  Electrical  Engineering,  vol.  i.,  p.  205,  vol.  ii.,'p.  677, 

1907. 


INSULATION  RESISTANCE  OF 
NETWORKS 


CHAPTER  VI 

Insulation  Resistance  of  Networks 

Insulation  resistance — Measuring  fault  and  insulation  resistance 
in  a  2-wire  system — 3-wire  system — Graphical  construction 
for  potentials — General  theorem — Measurement  of  insulation 
resistance — Example — Regulating  the  potential  of  the  mains 
— Leak  in  the  positive  outer — Leak  in  the  middle  main 
— Numerical  example — Energy  expended  in  earth  currents — 
Leakage  currents — Numerical  examples — The  values  of  /lf 
/2,  and  /3 — References. 

Insulation  THE  practically  universal  adoption  of  pressures 
of  supply  greater  than  200  volts  has  brought 
into  prominence  the  importance  of  knowing  the  insulation 
resistance  of  the  various  portions  into  which  a  network  of 
wires,  for  supplying  electric  power,  can  usually  be  divided. 
The  insulation  resistance  of  a  network  to  earth  is  defined  to 
be  the  resistance  between  all  the  conductors  of  the  network 
connected  in  parallel  and  the  earth.  In  this  chapter  we  shall 
describe  methods  of  measuring  this  resistance  and  we  shall 
also  show  how  a  knowledge  of  its  value  gives  us  important 
information  as  to  the  leakage  currents  and  consequent 
power  losses  in  the  network.  When  a  regular  record  is  made 
of  the  insulation  resistance  not  only  of  the  whole  network 
but  also  of  the  various  portions  of  it,  timely  notice  is  often 
given,  by  a  gradual  fall  in  the  value  of  the  resistance,  of  the 
development  of  a  fault.  This  fault  can  in  most  cases  be 

readily    located    by  the    methods   described    in  the   next 

111 


112        ELECTRIC   CABLES   AND   NETWORKS 

chapter,  and  rectified.  We  shall  first  describe  how  the  fault 
resistance  of  each  of  the  mains  of  a  2- wire  network  can  be 
found  by  means  of  a  voltmeter,  and  an  ammeter  or  a  resist- 
ance of  known  value.  In  some  cases  the  resistance  of  the 
voltmeter  itself  can  be  utilized  as  the  known  resistance. 

Fault  ket  P  anc^  ^  (-^8-  32)  denote  the  cross  sections 

resistance      Qf  tne  two  mamSj  the  pressure  V  between  which 


FIG.  32. 

is  kept  constant  by  means  of  a  dynamo  or  battery.  By  the 
fault  resistance  /i  of  the  positive  main  P,  we  denote  the  com- 
bined resistance  of  all  those  stray  paths  from  it  to  earth  along 
which  leakage  currents  flow,  and  similarly  /2  denotes  the 
resultant  resistance  of  the  paths,  in  the  insulating  materials 
used,  through  which  the  current  flows  from  the  earth  to  the 
negative  main  N.  We  do  not  consider  that  a  conductor  at 
zero  potential  belongs  to  "  earth  "  unless  it  is  in  good  elec- 
trical connexion  with  earth.  A  metallic  portion  of  a  switch, 
for  instance,  mounted  on  a  porcelain  base  may  be  at  zero 
potential,  and  yet,  the  resistance  of  any  stray  paths  from  it 
to  either  main  are  not  included  in  /i  or  /2 .  Similarly  any  part 
of  the  path  between  P  and  N  at  zero  potential  does  not  belong 
to  "  earth  "  unless  the  resistance  between  it  and  "  earth  " 
water  pipe  for  example — is  comparable  in  magnitude 


INSULATION  RESISTANCE  OF  NETWORKS     113 

with  the  joint  resistance  of  the  direct  paths  to  earth  from  P 
orN. 

In  practice  we  may  consider  that,  to  a  first  approximation, 
/!  and  /2  are  independent  of  the  load  between  P  and  N. 
When  we  switch  on  a  lamp  between  P  and  N,  a  portion  of  the 
connecting  wire  leading  to  the  lamp  is  added  on  to  the  posi- 
tive main.  This  portion  previously  to  switching  on  would 
be  at  the  same  potential  as  the  negative  main  and  would  be 
virtually  part  of  it.  If  there  was  a  leak  in  the  portion  of  it 
beyond  the  lamp,  we  see  that  when  the  switch  is  open  this  leak 
is  credited  to  the  negative  main  but  after  it  is  closed  to  the 
positive  main.  In  this  case  /i  and  /2  will  vary  with  the  load. 

If  FI  denote  the  potential  of  the  positive  main  we  shall 
assume  that  FI//I  gives  the  value  of  the  leakage  current  from 
this  main.  On  open  circuit,  this  assumption  is  admissible. 
On  a  heavy  load,  it  is  admissible  as  a  rough  approximation. 
If  the  voltage  drop  be  not  more  than  5  per  cent,  and  if 
the  service  circuits  are  well  insulated,  the  inaccuracy  intro- 
duced by  our  assumption  will  not,  in  the  great  majority  of 
cases,  be  greater  than  5  per  cent. 

The  insulation  resistance  F  of  the  network  is  given  by 

l/F  =  l/A  +  l//2 (1). 

Hence,  if  closing  a  switch  transfer  a  leaky  path  from  /2  to  /13 
the  value  of  F  is  unaltered. 

In    a    2-wire    system    when    a  voltmeter    is 

Measuring 

fault  and      available,  the  ratio  of  the  fault  resistances  of 

insulation 

resistance      the  two  mains  can  be  determined  immediately. 

in  a 

2-wire        We  shall  first  suppose  that  the  voltmeter  is  not 
system 

electrostatic  and  that  its  resistance  is  R.     When 

it  is  connected  between  P  (Fig.  32)  and  a  water  pipe  or  other 
good  earth  let  the  reading  be  F/.  Similarly  when  con- 
nected between  N  and  earth  let  it  read  F2'.  In  this  case 
Fs'  will  be  a  negative  quantity.  In  the  first  case,  since  by 


114        ELECTEIC  CABLES  AND  NETWOKKS 

KirchhofFs  law,  the  sum  of  the  currents  through  E  and  /! 
must  equal  the  current  flowing  from  the  earth  through  /2,  we 
have 

V1'/fi  +  Vl'/K=(V—  FO//2       ..   (2). 
In  the  second  case,  we  have 

-F27/2-F27^=(F+F2')//i     ..     (3). 
Hence  it  readily  follows  that 

A//2   -   -   -  Fi'/F,'       (4), 

/,   =   -  -  BUV—VS+Vt'W}     ..     (5), 
and  /2   =   R{(V—  Fi'+FO/Fi'}  ..     (6). 

From  (1),  (5),  and  (6),  we  see  also  that 

F=E{V/(Vi'—  F2')— 1}        ..      ..     (7). 

For  example,  suppose  that  the  resistance  R  of  the  volt- 
meter is  1,000  ohms,  that  F=  220,  Fi'  =  160  and  F2'  =  —  20 
volts,  respectively,  then 

by  (5),  /!  =  1,000 {(220— 160— 20)/20  }  =  2,000  ohms, 
by  (6),  /a  =  1,000  {(220— 160— 20)/160}  =  250  ohms, 
and  by  (7),  F  =  1,000(220/180— 1 }  =  222  ohms  nearly. 

Let  us  now  suppose  that  an  electrostatic  voltmeter  is  used, 
and  let  Fi  and  F2  be  the  potentials  of  the  two  mains  to 
earth  respectively.  In  this  case,  the  reading  of  the  volt- 
meter when  connected  between  the  positive  main  and  earth 
will  give  Fi  directly,  and  similarly  the  reading  between  the 
negative  main  and  earth  will  give — F2.  Equations  (2)  and 
(3)  may  now  be  written. 

Fi/A  =  (F— F1)//,=  F/(/1+/,)    -      ..     (8), 
and  -F2//2-(F+F2)//1=F/(/1+/2)   ..      ..     (9). 

These  equations  show  us  that  f±/f2  =  Fi/(  F — FI),  and  hence, 
when  F  is  known,  a  single  reading  FI  of  the  voltmeter  gives 
us  the  ratio  of  the  fault  resistances.  In  order  to  find  their 
absolute  values,  however,  further  measurements  must  be 
made.  For  example,  we  may  connect  between  the  positive 
main  P  and  earth  a  resistance  and  a  milli-ammeter  in  series. 


INSULATION  EESISTANCE  OF  NETWOEKS      115 

If  C  be  the  reading  of  the  ammeter  and  Ft'  the  new  read- 
ing of  the  electrostatic  voltmeter,  we  have,  by  Kirchhoff's 
law 


and  by  (8),  Vi/h=(V—Vi)/f*9 

and    thus,    subtracting,    (Fi—  FI')//I—  C=—  (V,—  F/)//2. 

Hence,  ^=  (V.—V^/C         ...          .     (10). 

From  (8)  and  (9),  we  also  have, 

fi=(—V/V2)F      ......     (11), 

and  fs=(V/V,)F  .........     (12). 

As  an  example,  let  us  suppose  that  V=  200,  Fi  =  150,  and 
F/  =  50  volts,  and  that  (7  equals  0-0010  of  an  ampere.  We 
find,  by  (10),  that 

^=(150—  50)/0-001  =100,000  ohms, 
and  by  (11)  and  (12),  that 

/!=  (200/50)  100000   =400,000  ohms  nearly, 
and  f2=  (200/150)100000=  133,000  ohms  nearly. 

By  (8)  and  (9),  we  see  that  the  power  expended  in 
the  leakage  currents  to  earth  F12//i  +  F32//2,  equals 
F2/(/i+/2).  Hence  any  diminution  in  the  value  of  /i+/2 
always  increases  the  power  loss  due  to  leakage  currents. 

Again,  since 


=  (l/F){  F!—  (*•//,)  F}*+F.V(/i+/.), 

we  see  that,  if  we  regard  FI  as  the  only  variable  quantity,  the 
expression  for  the  power  lost  has  its  minimum  value,  when 
Vi=(F/f2)V,  that  is,  when  Ohm's  law  is  obeyed.  We  con- 
clude therefore  that  if  the  potential  difference  between  the 
mains  be  maintained  constant,  then  as  the  fault  resistances 
vary,  the  potentials  of  the  mains  vary  always  in  such  a  way 
that  the  energy  expended  in  leakage  currents  is  a  minimum 
(see  Chapter  I). 


116        ELECTKIC  CABLES  AND  NETWORKS 


3-wire 
system 


We  shall  now  consider  how  the  potentials  of 
the  three  mains  in  a  3-wire  system  of  distribution 
vary  with  the  fault  resistances  of  the  three  mains.  As 
practically  all  direct  current  networks  are  supplied  on 
the  3-wire  system,  this  problem  is  one  of  considerable 
practical  importance. 

Let  P,  M,  and  N  (Fig.  33)  be  the  sections  of  the  positive, 


E 
FIG.  33. 

middle,  and  negative  mains  of  the  system  and  let  /l5  /2,  and  /3, 
be  the  fault  resistances  of  these  mains  respectively.  By  the 
fault  resistance  fi  we  denote  the  resultant  resistance  of  all 
the  leakage  paths  from  the  main  P  to  earth  which  do  not 
pass  through  the  main  M.  If  the  potential  of  the  main  M 
be  positive  and  there  are  lamps  switched  on  between  P  and 
M ,  it  is  obvious  that  there  will  be  leakage  paths  to  earth 
through  these  lamps  and  then  through  the  insulation  of  the 
main  M .  Even  when  there  is  no  load  between  P  and  M  we 
may  have  current  flowing  along  leakage  paths  from  P  and  M, 
and  then  to  earth.  It  has  to  be  remembered  that  these 
leakage  paths  directly  connecting  the  mains  and  insulated 
from  earth  are  not  included  in  the  fault  resistances  /i,  /2, 
and  /3.  These  values  merely  give  the  resultant  resistances 
of  the  direct  leakage  paths  to  earth  from  each  main. 


INSULATION  RESISTANCE  OF  NETWORKS     117 

The  insulation  resistance  F  of  the  network  is  defined  by 
the  equation 

l/F  =  1//1  +  1//2  +  1//3. 

It  is  therefore  the  insulation  resistance  between  the  three 
mains  in  parallel  and  the  earth.  F  generally  remains  ap- 
proximately constant  at  all  loads,  for  when  a  switch  is 
turned  on  between  the  positive  main  and  the  middle  main, 
for  instance,  some  of  the  leakage  paths  may  be  taken  from 
one  main  and  given  to  the  other,  but  usually  l//!+l//2 
remains  very  approximately  constant.  When,  however,  a 
double  pole  switch  is  used  for  a  leaky  subcircuit,  F  is 
diminished  when  the  switch  is  turned  on. 

In  Fig.  33  let  FI,  F2,  and  F3,  be  the  potentials  of  the  three 
mains  P,  M,  and  N.  Since  there  can  be  no  accumulation  of 
electricity  in  the  earth,  we  have,  by  Kirchhoff's  law, 

Fi//i  +  Fa//a  +  F3//3=0  ..  ..  (13). 
We  may  either  have  F2  and  F3  negative,  or  F3  alone  may 
be  negative.  At  the  supply  station  the  potential  differences 
between  the  mains  P  and  M  ,  and  between  M  and  N,  are  each 
maintained  constant  and  equal  to  F  (suppose).  Hence 


Substituting  for  Fi  and  F3  from  (14)  in  (13)  we  get  a  simple 
equation  from  which  F2  is  easily  found  in  terms  of  F,  /i,  /2, 
and  /3.  Hence  also,  from  (14),  we  find  FI  and  F3  in  terms  of 
these  quantities.  The  following  graphical  construction  is 
quite  as  simple  as  this  method  and  is  easier  to  apply  in 
practice. 

Draw  a  line  PN  (Fig.   34)  and  make  PM  — 

cSSon  MN=V-     Place    Particles   of   mass    I//,,    l//2, 

potentials      and  l/f3'  at  P>  M>  and  N>  respectively  and  let 

0  be  their  centre  of  gravity.     We  shall  consider 

that  lines  measured  in  the  direction  GP  are  positive  and  in 


118        ELECTRIC  CABLES   AND  NETWORKS 


the  direction  ON  negative.     Taking  moments  about  G,  we 

have 

0  ......     (15). 


H  —  I 
V 

/ 

—  vt  —                     *- 

a      V            K*         n 
G                        A/                                           /: 

I 

r 

_> 

i 

'                                    S 

FIG.   34.      Statical   diagram    illustrating  the    connexions  between    the 
potentials  and  the  fault  resistances  of  a  three-wire  distributing  system. 


(16). 


We  also  have 

OP=OM+V\ 

and  GN=GM—  V)       "      " 

Comparing  (15)  and  (16)  with  (13)  and  (14),  we  see  at  once 

that 

GP  =  Vi,  GM  =  V2  and  GN  =  V3. 

To  find  the  potentials  of  the  mains,  therefore,  when  the 
fault  resistances  /i,  /2,  and  /3,  are  known,  we  proceed  as 
follows  : — Choosing  a  suitable  scale  draw  a  straight  line  NP 
(Fig.  34)  to  represent  2V,  where  V  is  the  voltage  of  supply. 
Bisect  this  line  in  M,  and  find  the  centre  of  gravity  G  of 
masses  1//1?  l//2,  and  l//3,  placed  at  P,  M,  and  N  respectively. 
Then  the  potentials  of  the  three  mains  are  GP,  GM,  and 
GN  respectively. 

General  ^n  genera^'  ^  we  nave  n  mains  whose  fault 

theorem  resistances  are  /i,  /2,  /3,  . .  and  if  the  potential 
differences  between  them  are  V,  V,  V",  . .  the  potentials 
Fi,  F2,  Fa,  . .  of  the  mains  are  given  by  the  following 
construction.  Draw  a  straight  line  PiPn  the  length  of 
which  represents  F+F'-f-F"-}-  ..  .  Mark  the  points 
P2,  P3,  . .  on  it,  where  PiP2  =  V,  P2P3  =  V,  etc.  Place 
particles  of  mass  l//l5  l//2,  . .  at  Pl9  P2,  . .  respec- 


INSULATION  RESISTANCE   OP  NETWORKS    119 

lively,  and  let  G  be  their  centre  of  gravity.     Then  it  is 
easy  to  see  that 

V^P.G,  V2=P2G,     ..     . 

Measure-          -Let  us  suPPose  that  the  middle  main  is  con- 
futation    nected   with   earth   at   the   generating   station 
resistance      through  a  small  resistance  and  an  ammeter,  the 
reading  of  which  is  C.     Let  us  also  suppose  that  the  volt- 
meter connected  between  the  middle  main  and  earth  reads 
F2.     If  we  now  break  the  current  in  the  earth  circuit  so 
that  the  ammeter  reads  zero,  the  voltmeter  will  assume  a 
new  value  F2',  which  will  be  numerically  greater  than  F2. 
If  the  voltmeter  be  electrostatic,  the  insulation  resistance  F 
of  the  network  is  given  by 

F=(V2'—V2)/C (17). 

If  the  voltmeter  have  a  resistance  R,  we  obviously  have 
FK/(F+E)  =  (V2'—  V2)/C=F'  (suppose), 

and  thus,  F=F'R/(R—F') (18). 

In  either  case  the  insulation  resistance  is  found  almost  at 
once. 

We  may  prove  formula  (17)  as  follows.  Let  x  denote  the 
resistance  of  the  earth  connexion  with  the  middle  main,  and 
let  the  voltmeter  be  electrostatic.  Then,  by  Kirchhoff's  law, 
we  have, 

Fi//1+Fa//J  +  F,/*+F8//3  =  0      ..      ..     (19), 
and  F17/i  +  F,7/,+Fa7/3  =  0      ..      ..     (20). 

We  also  have 

Ft— F,  -  F2-F3  =  F, 

and  F/—  F2'  =  F2'—  F3'  =  F, 

and  therefore,  F/—  V,  =  V2'—V2  =  F3'— F3 ..     (21). 

Hence,  by  subtracting  (19)  from  (20),  we  get 

( F/-F1)//i+(  F2'-F2)//2+(  Fa'— Fa)//.  =  F2/*  =  C, 
and  therefore,  by  (21),   i//1  +  i//a  +  i//3  =  C7/(Fa'_F2), 
and  thus,  F=(V2'—V2)/C. 


120       ELECTRIC  CABLES  AND  NETWORKS 

When  the  voltmeter  has  a  resistance  R,  we  may  consider  that 
it  forms  one  of  the  leakage  paths  to  earth  on  the  middle 
main,  and  hence,  as  we  have  shown  above,  the  formula  can 
be  suitably  modified  without  difficulty, 

Let  us  suppose  that  initially  the  potential  of 

the  middle  main  was  8  volts,  and  that  the  reading 

on  the  ammeter  was  3-5  amperes.     Let  us  also  suppose  that 

when  the  earth  connexion  was   broken  the  voltmeter  read 

112.     Then,  if  the  voltmeter  is  electrostatic,  we  get  by  (17), 

F=(ll2—  8)/3-5=  29-7  ohms  nearly. 

If  the  voltmeter  had  a  resistance  of  400  ohms,  we  find,  by 
(18),  that 

^=29-7  x  400/(400—29-7) 

=  32-1  ohms. 

In  practice,  it  is  sometimes  more  convenient  to  connect 
the  positive  outer  through  a  resistance  and  an  ammeter  to 
earth.  The  earth  connexion  on  the  middle  main  being 
opened,  let  Fi  be  the  potential  of  the  positive  outer.  When 
the  switch  on  the  artificial  leak  on  the  positive  outer  is  closed, 
let  C  be  the  reading  of  the  ammeter  and  F/  the  new  reading 
of  the  voltmeter.  Then,  proceeding  as  before,  it  is  easy  to 
show  that 


The  maximum  pressure  of  supply,   between 
Regulating 

the          "  any  pair  of  terminals,"  to  the  ordinary  con- 
potentials  J  J 

of  the         sumer  is  fixed  by  the  Board  of  Trade  at  250  volts. 
mains 

The  object  of  this  regulation  is  to  prevent  shocks, 

at  pressures  greater  than  250  volts,  being  accidentally  re- 
ceived. If,  however,  the  absolute  value  of  the  potential  to 
earth  of  any  terminal  be  greater  than  250  volts,  it  is  obvious 
that  possible  shocks  can  be  obtained  between  this  terminal 
and  a  gas  or  water  pipe  or  a  damp  wall  or  floor.  To  carry  out 
the  object  of  the  regulation,  therefore,  it  is  necessary  to 


INSULATION  RESISTANCE   OF  NETWORKS   121 

prevent  the  potential  of  any  terminal  from  being  perma- 
nently greater  than  250  volts.  We  have  seen  above  that 
the  values  of  the  potentials  of  the  mains  depend  only  on 
the  pressure  maintained  between  them,  and  on  the  fault 
resistances.  The  graphical  construction  for  these  potentials 
(Fig.  34)  also  shows  us  that  by  making  a  large  artificial  leak 
on  the  middle  main,  so  that  l//2  is  large  compared  with 
either  l//t  or  l//3,  we  can  anchor  the  potential  of  the 
middle  main  so  that  it  never  differs  much  from  zero,  and 
so,  also,  that  the  potentials  of  the  positive  and  negative 
outers  never  differ  much  from  +  V  and — V  respectively, 
where  V  is  the  pressure  of  the  supply. 

It  is  found,  in  practice,  that  the  insulation  resistance  of 
the  negative  outer  of  a  3-wire  network  is  generally  much 
smaller  than  the  insulation  resistances  of  the  other  mains. 
The  flow  of  leakage  current  from  the  earth  seems  to  force 
moisture,  by  a  phenomenon  similar  to  endosmosis,  into  the 
insulating  covering  of  the  main,  and  thus  lowers  its  resistance. 
In  practice,  the  negative  outer  of  an  insulated  3-wire 
network  is  generally  at  a  small  negative  potential.  For 
example,  in  a  large  3-wire  system  in  London,  the  po- 
tentials of  the  mains  were  generally  about  190,  85,  and  — 20 
volts  from  earth  respectively  for  many  years.  If  the  voltage 
of  supply  had  been  doubled  the  potential  of  the  positive 
main  would  have  been  380  and  it  would  be  clearly  undesir- 
able to  have  parts  of  lampholders  and  switches  in  damp 
cellars,  etc.,  at  this  potential.  It  would  therefore  have  been 
necessary  to  prevent  the  potential  of  the  positive  outer  from 
exceeding  250  volts,  and  this  could  be  done  by  making  an 
artificial  leak  on  either  the  positive  outer  or  the  middle 
main.  We  shall  now  calculate  the  values  of  the  resistances 
of  the  leaks  which  would  be  necessary  in  order  to  reduce  the 
potential  of  the  positive  outer  to  a  given  value. 


122        ELECTRIC  CABLES  AND  NETWORKS 

Leak  Let  us  suppose  that   Fl5  F2,  and  F3,  are  the 

positive       potentials  of  the  three  mains,  and  that  F  is  the 

insulation  resistance  of  the  network  to  earth. 

Let  x  be  the  resistance  which  has  to  be  connected  between 

P  and  earth  in   order  to  reduce  its  potential    Fi  to  the 

required  value  F/.      If  C  be  the  current  in  the  leak  we 

have,  by  (17), 


but  G  is  also  equal  to  Vi'/x,  and  thus 

x  =  V,'F/(V,—V,/)      ......     (22). 

Leak  ^  we  eartn  the  middle  main  through  a  resist- 

Jm'ddfe      ance  V>  we  have 
main  F2'/</=  (  F2-F2')^, 

and  thus,  y=  F2'^/(Fa—  F2')      ..         ..     (23). 

It  has  to  be  remembered  that  Fi  —  Fi'  =  F2  —  F2',  and  thus 
the  current  in  the  earth  connexion,  (Fi  —  Vi)/F,  can  be 
predicted  beforehand. 

Numerical  ^et  us  suppose  that  initially  V±  =  300,  F2  =  100, 
and  V3  =  —  100  volts.  Let  us  also  suppose  that 
F  is  10  ohms.  We  shall  calculate  the  value  of  the  resistance 
x  which  has  to  be  placed  between  the  positive  outer  and  earth 
in  order  to  reduce  its  potential  to  250  volts.  We  have,  by 
(22), 

x  =  Vi'F/(  Fr-  F/)  =250  x  10/50  =50  ohms. 
The  current  in  this  leak  would  be  250/50,  that  is,  5  amperes. 
In  the  event  of  a  dead  earth  occurring  in  the  negative 
outer,  the  potential  of  P  would  be  nearly  400  volts,  and 
the  maximum  value  of  the  current  in  this  leak  would  be 
400/50,  that  is,  8  amperes. 

If  it  were  required  to  reduce  the  middle  main  M  to  zero 
potential,  the  value  of  x  would  be  given  by 
x=200x  10/100=20  ohms, 
the  ordinary  value  of  the  current  in  it  would  be  10  amperes, 


INSULATION  RESISTANCE   OF  NETWORKS    123 

and  the  maximum  value  of  this  current  would  be  20 
amperes. 

We  shall  now  calculate  the  value  of  the  resistance  y  which 
has  to  be  connected  between  the  middle  main  and  earth  in 
order  to  reduce  the  potential  of  the  positive  outer  to  250 
volts.  We  have,  by  (23) 

y  =  V2'F/(  F2-F2')  -  V*'F/(  Fi-F/) 

=  50x10/50=10  ohms. 

The  current  in  the  leak  would  be  the  same  as  in  the  preceding 
case,  namely  5  amperes,  and  as  the  maximum  possible  cur- 
rent occurs  in  it  when  either  of  the  outers  is  dead  earthed, 
we  see  that  the  maximum  possible  current  through  y  is  20 
amperes. 

If  the  resistance  of  y  were  zero,  F2'  would  also  be  zero  and 
thus  the  current  in  y  which  equals  (F2  —  Vz)/F  would  be 
(100—  0)/10,  that  is,  10  amperes. 

Energy  ^he    algebraical    expression  for    the    energy 

expended  in  earth  currents  is 


If  we  use  the  graphical  method,  shown  in  Fig.  34,  this  equals 


If  we  now  regard  the  position  of  G  as  variable,  by  a  well- 
known  statical  theorem,  this  expression  is  a  minimum  when 
G  is  the  centre  of  gravity  of  masses  1//1?  l//2,  and  l//3,  placed 
at  P,  M,  and  N,  respectively.  But  we  know  that  this  is  how 
the  potentials  adjust  themselves  in  practice,  and  hence  they 
adjust  themselves  so  that  the  energy  expended  is  a  minimum. 
An  analytical  proof  of  the  general  theorem  for  an  rc-wire 
system  can  be  given  as  follows.  If  we  suppose  that  x  is  the 
potential  of  the  positive  outer  P,  the  energy  expended  in 
leakage  currents  is 

*V/i+(»-F)VA+(*-F—  F')V/3+    -•      --     (24), 
where    F,    F',     .  .     are  the  potential  differences  between 


124         ELECTRIC   CABLES  AND  NETWORKS 

P!  and  P2,  P2  and  P3,  etc.     By  Kirchhoff's  law,  we  always 
have 

x/fi+(x—V)/fa+(x—V—V')/f*+  "  =  0  ..  (25). 
But  by  the  differential  calculus  this  is  the  equation  that 
determines  the  value  of  x  which  gives  to  the  expression  (24) 
a  maximum  or  a  minimum  value.  Since  the  second  dif- 
ferential coefficient  of  (24)  equals  2/F  it  is  always  positive 
and  hence  (24)  is  a  minimum  when  x  has  its  working  value 
which  is  given  by  (25). 

We  shall  now  consider  the  value  of  the  increase  in 
the  power  lost  due  to  earthing  the  middle  main.  We 
have 


If  we  connect  the  middle  main  directly  to  earth,  so  that 
F2  is  zero  and  V±  —  —  F3  =  F,  the  power  expended  in  earth 
leakage  currents  is  F2//i  +  F2//2.  Hence  F22/.F  is  the 
value  of  the  increase  in  the  power  loss  due  to  earthing  the 
middle  wire.  In  the  numerical  example  given  above,  F2  is 
100  and  P  is  10,  and  hence  we  see  that  the  loss  of  power 
entailed  by  earthing  the  middle  wire  would  be  a  kilowatt. 
In  some  of  the  older  100  volt  3-  wire  systems  in  England 
F  is  only  2  or  3  ohms,  and  there  have  been  cases  where 
it  has  been  less.  In  these  cases,  the  increase  in  the  loss  of 
power  due  to  compulsory  earthing  of  the  middle  wire  would 
be  appreciable. 

If  we  earth  the  middle  main  through  a  resistance  x,  the 
new  values  F/  and  F'  of  F2  and  F  are  given  by 


or  V2'  =  V2x/(F+x)  ..........    (26), 

and  F'  =  Fx/(F+x). 

The  increase  in  the  leakage  power,  therefore,  due  to  earthing 

the  middle  main  through  a  resistance  x9  equals 


INSULATION  RESISTANCE   OF   NETWORKS    125 

*  x{(F+x)/Fx} 


..........     (27). 

Leakage  -^  *s  important  to  notice  that  all  methods  of 

currents  preventing  the  potentials  of  the  mains  from 
rising  above  250  volts  increase  the  leakage  currents  to  earth. 
As  these  currents  are  always  flowing  it  is  desirable  to  keep 
them  as  small  as  possible  owing  to  the  electrolytic  damage 
they  may  do.  In  the  Board  of  Trade  conditions  for  the 
approval  of  earthing,  it  is  provided  that  a  record  of  the 
current  to  earth  through  the  earth  connexion  shall  be  kept, 
and  that  if  at  any  time  it  exceeds  the  one-thousandth  part 
of  the  maximum  current  of  supply,  immediate  steps  shall 
be  taken  to  improve  the  insulation  of  the  system.  Now 
the  current  that  is  measured  in  this  case  is  the  difference 
between  the  leakage  currents  from  the  positive  and  negative 
mains.  Even  when  the  fault  resistances  of  the  three  mains 
are  very  low,  yet  if  the  fault  resistances  of  the  two  outers  be 
nearly  equal,  the  current  in  the  earth  connexion  may  be 
very  small.  Hence  the  current  in  the  earth  connexion  is  no 
sure  guide  as  to  the  insulation  of  the  network.  A  better 
rule  is  to  insist  that  the  insulation  resistance  F  of  the 
network,  when  the  earth  connexion  is  removed,  is  always 
above  a  certain  value.  In  order  to  get  a  rough  idea  of  a 
suitable  minimum  value  for  the  insulation  resistance  of  a 
network,  we  shall  now  consider  the  eighth  of  the  Board 
of  Trade  regulations  (A).  This  regulation  is  as  follows  :  — 
"8.  Maintenance  of  Insulation.  —  The  insulation  of  every 
complete  circuit  used  for  the  supply  of  energy,  including 
all  machinery,  apparatus  and  devices  forming  part  of,  or 
in  connexion  with,  such  circuit,  shall  be  so  maintained  that 
the  leakage  current  shall  not  under  any  conditions  exceed 
one-thousandth  part  of  the  maximum  supply  current  ;  and 


126        ELECTRIC  CABLES   AND  NETWORKS 

suitable  means  shall  be  provided  for  the  immediate  indica- 
tion and  localization  of  leakage.  Every  leakage  shall  be 
remedied  without  delay. 

"  Every  such  circuit  shall  be  tested  for  insulation  at 
least  once  in  every  week,  and  the  undertakers  shall  duly 
record  the  results  of  the  testings. 

"  Provided  that  where  the  Board  of  Trade  have  approved 
of  any  part  of  any  electric  circuit  being  connected  with 
earth,  the  provisions  of  this  regulation  shall  not  apply  to 
that  circuit  so  long  as  the  connexion  with  earth  exists." 

For  a  2- wire  system  this  rule  is  clear.  It  proceeds  on 
the  assumption  that  the  permissible  leakage  power  must 
always  be  the  same  fraction  of  the  total  output.  In  other 
words,  if  we  double  the  pressure  of  supply,  the  output 
remaining  the  same,  the  lowest  permissible  value  of  the 
insulation  resistance  is  increased  four  times.  Hence,  from 
the  fire  risk  point  of  view,  the  various  2-wire  systems  of 
supply  are  treated  equitably.  The  systems,  however,  that 
use  lower  pressures  are  allowed  to  have  larger  leakage 
currents,  and  in  the  course  of  time  the  electrolytic  damage 
done  by  them  may  be  appreciable. 

When  We  try  to  apply  the  above  rule  to  3-wire  net- 
works we  are  met  with  the  difficulty  that  there  is  no  simple 
method  of  finding  /i,  /2,  and  /3,  and  hence  the  determination 
of  the  earth  leakage  currents  is  difficult.  We  can,  however, 
easily  determine  the  insulation  resistance  F  to  earth  of 
the  network  and  the  potentials  F1?  F2,  and  F3,  of  the  mains. 
The  question  therefore  arises,  having  given  the  potentials 
of  the  mains  and  the  insulation  resistance  of  the  network, 
can  we  determine  superior  limits  to  the  value  of  the  greatest 
leakage  current  from  a  main  and  to  the  value  of  the  leakage 
power  expended  in  the  earth  currents  from  the  mains. 
If  we  know  the  maximum  and  minimum  possible  values 


INSULATION  EESISTANCE   OF  NETWORKS    127 

of  these  quantities  for  given  values  of  F,  Fi,  F2,  and  F3, 
we  may  possibly  be  able  to  fix  on  a  minimum  permissible 
value  to  F  in  any  given  case.  It  would  of  course  be  pre- 
ferable to  determine  f1}  /2,  and  /3,  in  every  case,  but  the 
method  of  doing  this  described  below  is  difficult  and  small 
percentage  errors  in  the  readings  of  the  instruments  used 
may  lead  to  large  percentage  errors  in  the  values  of  the 
quantities  found. 

Let  us  now  suppose  that  the  values  of  F,  Fi,  F2,  F3,  and 
F,  are  known  and  investigate  the  maximum  value  of  the 
leakage  current  — F3//3  to  the  negative  main.  When  the 
middle  main  is  at  positive  potential,  which  is  the  usual 
case  in  practice,  the  leakage  current  to  the  negative  main 
being  equal  to  the  sum  of  the  leakage  currents  from  the 
positive  and  middle  main  will  be  numerically  the  greatest 
of  them,  and  hence  this  is  the  current  to  which  we  have  to 
find  the  superior  limit.  This  is  done  easily  from  the 
graphical  construction  given  in  Fig.  34.  In  the  problem 
considered,  G  is  fixed,  and  so  also  is  the  sum  l/F  of  the 
masses  placed  at  P,  M  and  N.  Now  — F3//3,  the  leakage 
current  to  the  negative  main,  is  the  moment  about  G  of 
the  mass  l//3  placed  at  N.  We  have  to  find,  therefore,  the 
maximum  value  of  this  moment  subject  to  the  condition 
that  the  sum  of  the  three  masses  at  P,  M  and  N  is  a  con- 
stant quantity  and  that  their  centre  of  gravity  G  is  a  fixed 
point.  If  the  mass  at  M  be  not  zero,  we  can  increase  the 
values  of  l//i  and  l//3,  by  dividing  this  mass  into  two  por- 
tions in  the  ratio  of  — F3  to  Fi,and  placing  these  portions 
at  P  and  N  respectively.  This  procedure  would  increase  the 
mass  at  N  without  altering  the  position  of  G  or  the  value 
of  F.  Hence  l//3  has  its  maximum  possible  value,  subject 
to  the  given  conditions,  when  l//2  is  zero.  In  this  case, 

^  =  Fi/A  = — F3//3. 


128        ELECTRIC  CABLES  AND  NETWORKS 

Hence        C^JV.-O^JV,  =  1/A+l//,  =  l/F, 
and  thus  (?„«.  =  {  ViV^Vi—V^/F 

=  {(V—Vt)(V+V,)/*V}/F       ..     ..     (a). 

This  is  also  the  maximum  possible  value  of  the  earth 
leakage  current  from  the  positive  outer,  and  it  is  greater 
than  any  possible  current  from  the  middle  main.  Hence 
(a)  fixes  a  superior  limit  to  the  value  of  the  earth  leakage 
currents. 

When  the  potentials  of  the  mains  and  the  insulation 
resistance  of  the  network  are  known,  proceeding  as  above, 
it  is  easy  to  show  that  the  smallest  possible  value  of 
the  greatest  earth  current  from  any  of  the  three  mains 
occurs  when  there  is  no  leakage  from  one  of  the  outers. 
Assuming  that  /i  is  infinite  we  find  that 

<?»„.  =  PV/2  =  -F3//3, 
and  hence,  proceeding  as  before,  we  get 

Cmi«={(V-V2)/F}(V,/V)  ......    (b). 

If  P  denote  the  power  expended  on  leakage  currents,  we 
have 


—  FV/2. 
It  is  therefore  a  maximum  when  /2  is  infinite,  and  so 

p      —(V2  _  V^}/F=2VC  (c) 

*  max.  —  \  *  r  2   )/  •*-    —  *  r  ^max.     ......        Vv/' 

The  minimum  value  of  P  occurs  when  l//2  has  its  greatest 
value,  that  is,  when  /i  is  infinite.  In  this  case  the  leakage 
current  between  the  middle  main  and  the  negative  outer 
is  Cmin,  and  therefore 

Pmin.  =  F,(  V-Vj/F  =  VG^n,  ......     («*). 

Let  us  now  suppose  that  the  middle  wire  of  this  system 
is  connected  with  earth  through  a  resistance  of  x  ohms.  If 
F2'  be  the  new  potential  of  the  middle  wire,  we  have,  by  (23), 


As  before  we  see  that  the  leakage  current  to  the  negative 


INSULATION  RESISTANCE   OF  NETWORKS    129 

or  from  the  positive  outer  will  have  its  maximum  value 
when  /2  is  infinite.  In  this  case  if  Cfmax%  denote  the  maximum 
value  of  the  current,  and  we  suppose  V2'  positive,  we  have 


Since  l/A  +  l//3=  1/F,  and  V,/h  =  —  F3//3,  we  get 
C'M  =  F/(  V—V,)/2VF+(  V2-V2')/F 

=  (V—V2')(V+V2)/2VF     ......   (a') 

and,  by  (26),  F2'  is  given  by 


Thus,  since  F2'  is  always  less  than  F2,  C'^^^  is  always  greater 
than  CmaXf  Earthing  the  middle  main  through  a  resistance, 
therefore,  always  increases  the  superior  limit  of  the  possible 
values  of  the  leakage  currents. 

Similarly   the    minimum    value     C"min    of    the    leakage 
current  occurs  when  /t  is  infinite,  and  hence 


Comparing  this  with  (b)  we  see  that  C1  'min.  is  greater  than 

<u. 

If  P'max.  denote  the  maximum  possible  value  of  the  power 

expended  on  leakage  currents,  it  is  not  difficult  to  show  that 

P'mm=V*/F-{x/(F+x)}(VS/F)      ..     ..  (c'). 

Similarly  we  can  show  that 

P'min=VV*/F-{x/(F+x)}(VS/F)    ..     ..(«*'). 

It   follows   by   comparing   the   corresponding     formulae 
that  P'moa;.  and  P'TOin.  are  respectively   greater  than  Pmaa. 
and  Pmin>. 

In  the  particular  case,  when  the  middle  main  is  dead 
earthed   so  that   both  x  and   F2'  are  zero,  the   formulae 
become 

.',..    ..      ..     (a*), 

(6"), 
(c"), 
and  P"A  =  VV*/F       ........      (d"). 

K 


130        ELECTRIC  CABLES  AND  NETWORKS 

The  above  formulae  show  that  whether  we  earth  the 
middle  wire  or  not,  (F+F2)/2^,  and  a  fortiori  V/F,  is  a 
superior  limit  to  the  value  of  the  earth  current  to  or  from 
any  main.  They  also  show  that  V2/F  is  a  superior  limit 
to  the  power  expended  in  earth  currents. 

For  instance,  if  V  =  220  volts,  and  F  =  25  ohms,  the  power 
expended  in  leakage  currents  cannot  be  greater  than 
2202/25,  that  is,  1-936  kilowatts,  and  no  earth  current  can 
be  greater  than  V/F,  that  is,  8-8  amperes.  If  the  values 
of  F2  and  x  be  known,  we  can  in  general  reduce  these  values 
considerably.  The  following  numerical  examples  illustrate 
how  readily  the  above  formulae,  which  are  due  to  the  author, 
can  be  applied  in  practice. 

Numerical         -^et  us   suppose   that  the   maximum   output 

examples  of  a  3_wire  djrect  current  station  with  400 
volts  between  the  outers  is  3,000  kws.  We  shall  calculate 
the  lowest  insulation  resistance  which  will  ensure  that  no 
earth  leakage  current  is  greater  than  the  thousandth  part 
of  the  maximum  supply  current,  the  potential  of  the  middle 
main  being  40  volts. 

In  this  case  F=200,  F2=40,  the  maximum  current  of 
supply  is  (3,000,000/400),  that  is  7,500  amperes,  and 
therefore  the  maximum  leakage  current  must  not  exceed 
7*5  amperes.  Substituting  these  values  in  (a),  we  get 

7-5  ={  (200— 40) /F }  { (200+40)/400 } , 

and  therefore  -F  =  12'8  ohms.  Hence  if  the  insulation  resist- 
ance of  the  network  be  greater  than  12-8  ohms  the  maximum 
value  of  the  leakage  current  from  any  part  of  the  three 
mains  will  be  less  than  the  thousandth  part  of  the  maximum 
supply  current. 

The  maximum  possible  value  of  the  power  expended  in 
the  currents  to  earth  in  this  case  is,  by  (c), 

— 402)/12'8=3 


INSULATION  RESISTANCE   OF  NETWORKS    131 

If  the  middle  wire  had  to  be  dead  earthed  we  see,  by  (a"), 
that 

F=IQ  ohms, 
and  by  (c"), 

PmaXf  =  2002/12-8=  3-125  kws. 

Let  us  now  suppose  that  the  pressure  between  the  outer 
mains  was  reduced  to  200  volts,  so  that  F  — 100,  and  V2  =20. 
If  the  output  of  the  station  remained  the  same  the  maxi- 
mum permissible  leakage  current  would  be  15  amperes. 
In  this  case  for  the  insulated  network,  by  (a),  F  must  not 
be  less  than  {(100— 20)/15}  {(100+ 20)/200},  that  is, 
3 '2  ohms,  and  for  the  earthed  network,  F  must  not  be 
less  than  4  ohms.  The  values  of  the  leakage  power  ex- 
pended in  the  leakage  paths  would  be  as  before  3  and 
3-125  kws. 

In  practice,  it  is  not  permissible  to  have  a  voltage  drop 
in  the  mains  greater  than  4  per  cent.,  and  hence  (Chapter  V) 
the  maximum  load  on  a  low  pressure  network  varies  as  the 
square  of  the  voltage.  The  maximum  permissible  load, 
therefore,  when  the  pressure  is  halved  is  only  one-quarter 
of  its  original  value,  and  thus  the  maximum  permissible 
leakage  current  is  only  3-75  amperes,  and  the  value  of  F, 
therefore,  is  16  ohms,  the  same  value  as  before. 

As  a  further  example,  let  us  suppose  that  the  potentials 
of  the  mains  of  a  3-wire  direct  current  system  are  300,  100, 
and  — 100  volts,  respectively.  Let  us  also  suppose  that  the 
insulation  resistance  F  of  the  system  is  found  to  be  10  ohms. 
We  shall  find  the  limits  between  which  the  greatest  of  the 
earth  currents  must  lie  and  also  the  limits  between  which 
the  leakage  powrer  must  lie,  both  when  the  network  is 
insulated  and  when  the  middle  wire  is  connected  with  earth 
through  a  resistance  of  2  ohms. 

By  formulae  (a),  (6),  (c),  and  (d),  we  at  once  find  that 


132        ELECTRIC  CABLES  AND  NETWORKS 

Cm(KC  =(200— 100)(200+100)/400  x  10  =7-5  amperes, 
<7miri  =(200— 100)100/200  xlO  =5  amperes, 

P^  =400  x7-5  =3  kws., 

and  Pmin.  =200x5  =1  kw. 

When  the  resistance  x  of  the  earth  connexion  of  the 
middle  main  is  2  ohms,  we  have,  by  (26), 

V2'={x/(F+x)}V2=  100/6=  16-7  volts  approx., 
and  therefore,  F/  =  216-7  and  V3'=— 183-3  volts. 
We  easily  find  by  (a7),  (&'),  (O>  and  (d'),  that 

C'max.  =  (200—16-7)300/400  x  10         =  13-75  amperes, 
0'mfci.  =  (200—16-7)100/200  x  10         =  9-17  amperes, 
P'waa,  =  2002/10— (1/6)1002/10  =3-83  kws., 

and  P'min<  =  200  x  100/10— (1/6)1002/10  =  1-83  kws. 
In  this  case,  the  current  in  the  earth  connexion  is  16-7/2, 
that  is  8-35  amperes. 

If  finally  we  suppose  that  the  middle  main  is  dead  earthed 
so  that  x  is  zero,  we  have  by  (a"),  (&"),  (c"),  and  (d"), 
C"m(KK.=  (200+100)/2  x  10=  15  amperes, 
C"'m.n  —  100/10  =  10  amperes, 

^"ma*.  =  2002/10  =4  kws., 

and,  P"min.  =  200  x  100/10          =  2  kws. 

We  could  have  predicted  at  once  that  the  maximum 
leakage  current  would  in  any  case  have  been  less  than 
V/F,  that  is,  20  amperes  and  that  the  maximum  leakage 
power  could  not  have  been  greater  than  V2/F,  that  is, 
4  kilowatts.  The  more  complicated  formulae,  however, 
give  us  valuable  additional  information. 

As  a  final  example,  we  shall  take  the  values  obtained  by 
measurements  made  in  1900  on  a  large  supply  network  in 
London.  In  this  case 

Ft=190,   F2=85,   F3=— 20,  and  P=2-5. 
We  shall  find  the  limits  between  which  the  maximum 
value  of  the  earth  current  to  any  of  the  mains  must  lie, 


INSULATION  RESISTANCE   OF  NETWORKS    133 

and  also  the  limits  for  the  leakage  power.  By  formulae 
(a)  and  (b),  we  have, 

OmaB.=(105— 85)(105+85)/210  x2-5=7-24  amperes, 
and  <7min  =85  X20/105  x2*5  =6-48  amperes. 

Whatever  may  have  been  the  actual  values  of  the  fault 
resistances  of  the  mains,  the  value  of  the  leakage  current 
to  the  negative  main  cannot  have  been  less  than  6-48  am- 
peres or  greater  than  7-24  amperes.  Similarly,  by  (c)  and 
(d),  we  find  that  the  value  of  the  leakage  power  cannot  have 
been  less  than  0*68  kw.  or  greater  than  1-52  kws. 

If  the  middle  main  of  this  network  had  been  earthed 
the  current  to  the  negative  main  would  have  had  some 
value  between  34  and  38  amperes,  and  the  power  expended 
in  leakage  currents  would  have  had  a  value  between  3-57 
and  4-41  kws.  The  current  in  the  earth  connexion 
also  would  have  been  34  amperes.  In  this  case,  the  only 
advantage  gained  by  earthing  the  middle  wire  would  have 
been  the  reduction  of  the  potential  of  the  positive  main 
from  190  to  105  volts.  On  the  other  hand,  the  leakage 
current  to  the  negative  would  now  have  been  doing  five 
times  the  amount  of  electrolytic  damage,  and  in  order  to 
maintain  the  balance  of  the  potentials  about  3  kws.  would 
have  to  be  expended  in  the  leakage  paths  all  the  year 
round. 

These  examples  illustrate  that  a  knowledge  of  V2  and  F 
gives  us  most  useful  information  about  the  leakage  currents 
and  the  leakage  power  in  a  3- wire  network. 

j  There  is  no  good  practical  method  of  determin- 

of  A,  /2,     ing  the  values  of  /i,  /2,and  /3.     When  it  is  per- 
missible to  arrange  that,  during  the  brief  time 
required  to  take  the  necessary  readings,  the  voltage  between 
the  positive  and  the  middle  may  be  10  or  preferably  20  per 
cent,  different  from  the  voltage  between  the  negative  and 


134        ELECTRIC  CABLES   AND   NETWORKS 

the  middle,  the  following  method  will  give  approximate 
values  of  the  three  fault  resistances. 

Measure  in  the  ordinary  way,  first  of  all,  F,  F1?  F2  and 
F3.  This  gives  us  the  two  equations, 

1//1  +  1//2  +  1//3  =l/F  .....          (1), 

and  F1//1  +  F2//2  +  73/3  =  0     ......     (2). 

Next  upset  the  balance  of  the  pressures  so  that  the 
voltages  between  the  two  sides  of  the  3-wire  supply 
are  appreciably  different,  and  measure  the  new  values 
Fi',  F/,  and  F3',  of  the  potentials  of  the  mains. 

This  gives  us  the  further  equation  —  - 

Ft7A+F,7/>+F87/3  =  o     ......    (3). 

Hence  we  have  three  equations  to  determine  three  unknown 
quantities  l//l5  l//2,  and  l//3,  and  thus  solving  by  deter- 
minants, or  otherwise,  we  get 


and     l/A-tFiF/—  F2 

where    A=   F2F37—  F3F2/+F3F1/—  F1F3/+F1F2/—  F2F/. 

The  solution  shows  that  a  small  percentage  error  in  the 
value  of  a  voltmeter  reading  may  sometimes  introduce  a 
large  error  in  the  value  of  the  fault  resistances  deduced  by 
the  formulae.  Suppose,  for  instance,  that  FiF2'  —  F2Fi'  is 
12  and  that  FiF2'  is  2,000,  then  a  1  per  cent,  error  in  the 
reading  of  either  Fi  or  F2'  could  give  to  /3  an  impossible 
negative  value.  Hence  this  method  has  to  be  used  with 
caution. 

It  is  to  be  noticed  that  it  is  necessary  to  upset  the  balance 
of  the  voltages  in  order  to  get  equation  (3).  If  we  merely 
make  an  artificial  leak  in  one  of  the  mains  we  get 


where  C  is  the  current  in  the  leak.     But  since   Fi  —  Fi"  '  = 
F3—  F2"  =  F3—  F3",  and  F=  (V.—  V^/C,  equation  (4)  can 


INSULATION  RESISTANCE   OF  NETWORKS    135 

be  deduced  from  (1)  and  (2)  and  is  therefore  not  an  inde- 
pendent equation. 

REFERENCES. 

A.  Russell,  "  The  Regulation  of  the  Potentials  to  Earth  of  Direct 
Current  Mains."  Journ.  Inst.  EL  Eng.,  vol.  30,  p.  326,  1900. 

A.  M.  Taylor.  "  Network  Tests  and  Station  Earthing."  Journ. 
Inst.  EL  Eng.,  vol.  32,  p.  872,  1903. 

W.  E.  Groves.  "  Localization  of  Faults  on  Low- tension  Networks," 
Journ.  Inst.  EL  Eng.,  vol.  33,  p.  1029,  1904. 


FAULTS  IN  NETWORKS 


CHAPTER    VII 

Faults   in    Networks 

Faults  in  networks  —  House  wiring  —  Earths  —  Short  circuits  —  Breaks 
—Distributing  networks—  The  localization  of  faults—  Detecting 
faulty  mains  by  flashing  —  General  methods  —  The  Hopkinson 
3-ammeter  method  —  The  final  methods  of  localization  — 
The  fall  of  potential  method  —  Loop  test  —  2-ammeter  method 
—  Induction  method  —  Blavier's  test  —  Example  —  References. 


Faults  in  ^HE  f  au^s  that  most  commonly  arise  in  practice 
networks  are  (jue  f.Q  causes  which  can  be  roughly  classi- 
fied under  three  headings,  short-circuits,  earths,  and  breaks. 
A  short  circuit,  or  as  it  is  generally  called  in  America,  a 
"  cross,"  occurs  when  two  conductors  of  opposite  polarity 
get  connected  by  a  path  of  very  small  resistance.  The 
consequent  dangers,  of  fire  and  of  the  dynamos  being  over- 
loaded, arising  from  this  type  of  fault,  are  obviated  in 
practice  by  means  of  fuses  or  automatic  cut-outs.  An 
earth,  or  a  "  ground  "  as  it  is  sometimes  called,  occurs 
when  any  conductor  of  the  network  makes  contact  through 
a  path  of  small  resistance  with  the  earth.  Water  pipes, 
for  instance,  make  effective  contact  with  the  earth, 
and  if  a  metal  conductor  touch  a  water  pipe  in  such  a 
way  that  the  contact  resistance  is  very  small  it  makes 
what  is  called  a  "  dead  earth."  Sometimes,  however,  the 
resistance  of  the  fault  is  appreciable  and  we  get  what 
is  called  a  partial  earth.  If  systematic  tests  of  the  in- 
sulation resistance  to  earth  of  the  wiring  be  not  made 

139 


140        ELECTRIC  CABLES   AND  NETWORKS 

periodically,  this  kind  of  fault  may  remain  undetected 
for  a  long  time,  and  in  the  event  of  a  fault  developing 
on  a  main  of  opposite  polarity  there  will  be  a  risk  of 
fire.  The  third  kind  of  fault  occurs  when  there  is  a  partial 
or  a  total  break  in  a  conductor.  It  may  arise  owing  to  a 
terminal  screw  working  loose,  and  the  end  of  the  conductor 
ceasing  to  make  contact,  or  it  may  be  due  to  an  actual  break 
in  the  conductor  itself. 

The  localization  of  faults  in  a  distributing  network  is  an 
operation  demanding  not  only  considerable  skill,  but  also 
a  tJiorough  knowledge  of  how  the  cables  and  feeders  are 
arranged  in  the  network.  A  detailed  plan  of  the  wiring 
is  therefore  almost  essential,  and  ought  always  to  be  readily 
available.  Having  access  to  this  plan,  it  is,  as  a  rule,  not 
difficult  to  devise  a  method  of  procedure  which  must  ulti- 
mately locate  the  fault.  It  is  always  best  to  make  the 
search  in  a  methodical  and  thorough  manner.  The  youth- 
ful engineer,  for  instance,  sometimes  neglects  to  test  part 
of  a  section  simply  because  it  is  easier  to  disconnect  and 
make  rough  tests  at  the  sectional  pillars  than  at  the  under- 
ground manholes.  Hence  a  partial  earth,  which  could 
easily  be  detected  at  a  manhole,  may  be  left  undetected 
for  months. 

If  the  search  be  made  methodically  the  fault  or  faults 
cannot  fail  to  be  discovered.  Sometimes  the  first  test 
indicates  the  position  of  the  only  fault,  and  sometimes  the 
faulty  section  is  only  found  after  having  isolated  and 
tested  all  the  others. 

House  ^s    an    introduction    to    the    more    difficult 

wiring  case  o£  a  distributing  network  let  us  consider 
the  method  of  testing  for  faults  in  a  house  wiring  circuit. 
To  simplify  the  problem,  we  shall  consider  the  case  of  a 
house  installed  on  the  2-wire  system. 


FAULTS   IN  NETWORKS 


141 


Earths 


Let  us  first  suppose  that  the  insulation  re- 
sistance to  earth  of  the  wiring  shown  in  the 
diagram  (Fig.  35)  has  been  found  to  be  below  the  standard. 
We  have  therefore  to  locate  the  section  on  which  the 
partial  or  dead  earth  is  situated.  In  Fig.  35,  CMF  re- 
presents the  company's  main  fuse,  M  the  meter,  MS  the 
main  switch,  M F  the  main  fuse,  and  MDB  the  main  dis- 


FIG.   35. — Diagram  of  House  Wiring. 

tributing  board.     The  distributing  boards  for  the  various 
floors  are  marked  DB. 

We  shall  suppose  that  the  insulation  test  (see  p.  100) 
to  earth  has  been  made  at  the  company's  main  fuse  CMF, 
all  the  lamps  being  in  their  sockets  and  all  the  switches 
being  closed.  We  first  open  the  double  pole  switch  MS,  and 
repeat  the  test.  If  the  testing  instrument,  ohmmeter  let 


142        ELECTRIC  CABLES   AND  NETWORKS 

us  suppose,  now  read  infinity  we  see  that  the  fault  is  not 
in  connexion  with  the  meter  or  the  main  switch.  If, 
however,  it  reads  practically  the  same  as  when  the  switch 
was  closed,  the  fault  is  in  the  meter  or  the  base  of  the 
switch.  In  the  former  case,  a  bare  wire  is  probably  making 
contact  with  the  meter  cover,  and  in  the  latter,  the  base 
of  a  switch  may  be  a  conductor.  The  slate,  for  instance, 
of  which  it  is  made  may  have  metallic  veins.  This  can 
generally  be  remedied  by  bushing  the  fixing  screws  with 
ebonite. 

We  have  next  to  test  the  mains  on  the  house  side  of 
the  main  switch  MS.  Turn  off  the  switches  on  the  main 
distributing  board,  one  by  one,  and  take  the  reading  of  the 
ohmmeter  between  each  operation.  Let  us  suppose,  for 
example,  that  when  the  fourth  switch  is  turned  off  the 
reading  changes  very  appreciably.  In  this  case,  the  fourth 
switch  obviously  controls  a  faulty  section.  By  discon- 
necting the  leads  from  the  distributing  board  on  No.  4 
circuit,  we  can  readily  test  at  the  main  switchboard  whether 
the  fault  be  in  the  mains  connecting  the  two  boards.  If 
these  mains  be  free  from  faults  we  next  proceed  to  No.  4 
distributing  board,  and  test  it  in  the  same  way  as  the  main 
switchboard.  We  thus  finally  locate  the  faulty  lamp 
circuit.  The  fault  may  be  due  to  a  defective  switch  having 
been  placed  on  damp  plaster  or  an  abrasion  of  the  cover- 
ing of  one  of  the  wires  or  flexibles  may  provide  a  path  of 
small  resistance  to  a  neighbouring  gaspipe  or  a  steel  girder. 
Faults  are  often  found  also  in  ceiling  roses  or  lamp  brackets. 

Suppose  that  when  all  the  switches  are  turned  off  on 
the  main  switchboard  the  insulation  resistance  measured 
from  MS  still  reads  very  low.  "In  this  case  we  first  re- 
move the  fuses  in  MF,  and  test  the  insulation  resistance 
of  the  section  between  MS  and  MF,  and  of  the  base  of 


FAULTS   IN*  NETWORKS  143 

MF.     If  these  resistances  were  satisfactory  the  fault  or 
faults  must  be  on  the  main  switchboard. 

To  test  the  main  switchboard  we  disconnect  all  the 
mains  from  it.  We  then  join  all  the  metallic  parts  on  the 
face  of  the  board  with  binding  wire,  and  measure  the  in- 
sulation resistance  between  this  wire  and  earth.  By  dis- 
connecting the  binding  wire  from  each  of  the  metallic 
portions  in  turn,  and  reading  the  ohmmeter  between  each 
operation,  we  test  each  portion,  and  thus  readily  locate 
the  fault  or  faults.  Slate  of  inferior  quality,  badly  insu- 
lated from  the  fixing  screws  by  defective  bushing,  may 
easily  develop  bad  earth  faults. 


Short  -  e  seen  that  to  locate  an  earth  fault 

circuits  rapidly,  more  especially  when  its  resistance  is 
high,  an  ohmmeter  or  some  other  suitable  testing  instru- 
ment is  necessary.  The  location  of  a  short  circuit,  how- 
ever, requires  no  instrument  and  is  usually  exceedingly 
simple.  The  blowing  of  the  fuse  generally  locates  the 
faulty  circuit.  We  have  then  to  examine  the  lamp,  holder, 
ceiling  rose,  and  flexible  cord  to  find  out  where  contact 
between  conductors  of  opposite  polarity  is  taking  place. 

In  ordinary  installations,  short  circuits  can  occur  in  the 
lamp  holders,  and  in  the  flexible  wires  used  to  support  the 
lamps.  In  these  cases  the  fuse  protecting  the  circuit 
generally  blows  at  once,  and  thus  they  are  not  dangerous. 
When,  however,  a  flexible  wiring  system  is  used,  or  when 
a  switch  is  connected  with  flexible  wires,  a  more  dangerous 
partial  short  circuit  can  occur.  Let  us  suppose,  for  ex- 
ample, that  the  switch  S  (Fig.  36)  controls  the  lamps  L. 
If  a  short  circuit  occurs  at  A  or  (7,  the  fuse  will  immedi- 
ately blow,  but  if  the  short  circuit  occurs  at  B,  between 
the  wires  connected  with  the  switch,  the  lamps  will  still  be 
in  circuit.  Hence,  although  an  arc  will  start  at  J5,  the  fuses 


144        ELECTRIC  CABLES  AND  NETWORKS 

will  not  blow.  In  this  case  the  arc  will  probably  move 
slowly  up  the  flexible  until  the  mains  are  involved,  when 
the  fuse  will  almost  certainly  blow.  The  risk  of  fire  will 


MAINS 


-    :          i     HH 

!  LI 

1                   L 

D.PF. 

B_ 

- 

\ 

FIG.  36. — A  Short  Circuit  at  B  is  dangerous  in  flexible  Wiring  Systems. 

therefore  be  much  greater  when  the  short  occurs  at  B,  than 
when  it  occurs  at  either  A  or  C.  To  obviate  this  risk  a 
safety  device,  due  to  Coninx,  is  sometimes  employed.  A 
third  wire  (Fig.  37)  connected  with  the  opposite  main  is 

A  C 


MAINS 


HH 

DRF 

i                    L 

B 

0 

"")„ 

FIG.  37.— The  Safety  Wire. 

twisted  with  the  flexible  required  for  the  switch  connexion 
S.  Hence  in  the  event  of  an  arc  occurring  on  a  switch 
flexible,  a  dead  short  will  be  sure  to  occur  very  quickly 


FAULTS  IN  NETWORKS  145 

between  the  mains.  This  will  blow  the  fuses  before  the 
arc  has  time  to  set  fire  to  neighbouring  objects,  and  so  the 
fire  risk  will  be  minimized. 

A  break  in  the  continuity  of  the  conductors 
Breaks  J 

is  generally  easily  located  when  a  portable  volt- 
meter is  available.  If  the  switch  be  turned  on  we  can 
find  whether  two  parts  of  a  conductor  are  of  opposite  polarity 
by  noticing  the  reading  of  the  voltmeter  when  its  terminals 
are  connected  by  means  of  suitable  flexible  conductors 
across  the  two  parts.  If  no  pressure  be  indicated,  they 
are  both  on  the  same  side  of  the  break,  but  if  the  full  pres- 
sure be  indicated  they  are  on  opposite  sides  of  the  break. 
By  pushing  needles  through  the  insulation,  contact  can  be 
made  with  the  conductor  and  the  exact  position  of  the 
break  can  often  in  this  way  be  rapidly  located. 

Methods   of  rapidly   finding   the  position  of 

Distribut- 

i        ing         faults   on  a   distributing  network  are   of  con- 
networks        .IT,.  i 

siderable   importance   to   the   station   engineer. 

In  ordinary  low-tension  networks,  the  location  is  only  diffi- 
cult, when  the  network  is  closely  netted  by  numerous 
feeders.  Attention  is  often  directed  to  the  fault  at  once 
by  the  complaints  of  consumers,  and  blown  fuses  in  the 
manhole  section  boxes  or  the  section  pillars  indicate  the 
faulty  section. 

Let  us  consider  the  case  of  a  3- wire  low-tension  network 
with  its  neutral  earthed  through  a  resistance  of  2  ohms. 
In  order  that  faults  may  be  detected  as  soon  as  possible 
it  is  essential  that  daily  tests  of  the  insulation  resistance 
of  the  whole  system  to  earth  be  made.  The  chart  of  the 
recording  ammeter  in  the  earth  connexion  should  also  be 
closely  studied  to  see  if  there  is  any  periodic  variation. 
If  there  is,  it  is  probably  due  to  a  fault  in  a  private  instal- 
lation periodically  brought  in  and  cut  out  of  the  network 

L 


146        ELECTRIC  CABLES  AND  NETWORKS 

by  the  consumer's  switch.  A  continuous  rapid  oscillation 
of  the  ammeter  pointer  indicates  that  there  is  a  defective 
motor  armature  in  the  circuit.  If  the  ammeter  in  the 
earth  connexion  be  polarized  so  that  it  indicates  the  direction 
as  well  as  the  magnitude  of  the  current,  we  can  tell  at  once 
which  of  the  outers  has  the  greater  fault  resistance.  If 
the  current  be  flowing  from  the  middle  conductor  to  earth 
the  negative  outer  has  the  lower  fault  resistance,  but  if  it 
flow  in  the  other  direction  the  positive  outer  has  the  lower 
fault  resistance. 

When  the  reading  of  the  recording  ammeter  in  the  earth 
connexion  is  very  small,  it  has  to  be  particularly  noticed 
that  the  insulation  resistance  is  not  necessarily  high.  If 
the  fault  resistance  of  the  middle  main  be  very  low  or  if 
the  fault  resistances  of  the  two  outers  be  nearly  balanced, 
no  matter  how  low  they  may  be,  the  reading  of  the  recording 
earth  ammeter  may  be  very  small.  A  small  reading  of 
this  instrument  gives  no  certain  indication  of  the  magni- 
tude of  the  earth  faults  on  the  system,  but  a  large  reading 
indicates  that  the  insulation  resistance  of  at  least  one  of 
the  mains  is  very  low. 

The  daily  test  of  the  insulation  resistance  of  the  network 
(see  Chapter  VI)  gives  much  more  information  about  the 
magnitude  of  the  faults  than  the  readings  of  the  earth 
ammeter.  A  sudden  fall  in  the  value  of  this  quantity 
indicates  that  at  least  one  fault  has  suddenly  developed 
in  the  network.  An  inspection  of  the  chart  of  the  record- 
ing ammeter  may  indicate  the  exact  time  at  which  this 
fault  developed,  and  the  direction  of  the  flow  of  current  in 
the  earth  connexion  in  the  case  of  serious  faults  usually 
indicates  the  main  in  which  the  fault  has  taken  place.  If 
the  earth  ammeter  does  not  indicate  the  direction  of  the 
current  it  can  be  readily  found  by  Ampere's  rule  with  the 


FAULTS  IN  NETWORKS 


147 


help  of  a  small  compass.  It  is  advisable  to  leave  this  small 
compass  permanently  in  position  so  that  the  direction  can 
always  be  ascertained  by  a  glance.  The  polarity  of  the 
middle  main  with  reference  to  the  earth  can  also  be  easily 
found  by  pole  testing  paper. 

Having  found  the  outer  on  which  the  fault  exists  (the 
negative  suppose),  we  increase  the  resistance  in  the  earth 
connexion,  and  momentarily  close  an  artificial  leak  of  small 
resistance  in  the  sound  outer.  If  the  fault  exist  on  a 


/ 

'  ss 

RA 

r  E 

^  L 

s>  \ 

© 

^1 

~\       r 

~\ 

FIG.   38.— Earth  on  the  Middle  Wire. 


consumer's  circuit,  the  fuse  in  series  with  it  will  blow,  and 
his  complaints  will  determine  the  position  of  the  fault.  If 
closing  momentarily  the  artificial  leak,  that  is,  if  flashing 
does  not  clear  the  fault,  it  must  lie  on  a  part  of  the  main 
protected  by  a  large  fuse,  or  there  must  also  be  a  large 
fault  on  the  middle  main. 

us  suPPose  that  a  central  zero  ammeter 
8)  or  a  current  direction  indicating  am- 
meter  can,  by  opening  the  switch  L  or  by  taking 


Detecting 


flashing 


UNIVERSITY 

O .        Or 


148        ELECTRIC  CABLES  AND  NETWORKS 

out  a  plug,  be  readily  put  in  series  with  the  middle  main. 
Choose  some  time  of  the  day  when  the  load  is  very  small, 
and  put  the  ammeter  in  circuit  by  opening  L.  Increase 
also  the  resistance  of  the  earth  connexion  S2E,  or  preferably, 
if  permissible,  open  the  switch  S2  of  this  connexion.  Now 
flash  the  positive  main  P,  by  closing  the  switch  S\,  and 
notice  whether  this  operation  produces  a  throw  of  the 
ammeter  pointer.  If  it  does  there  must  be  a  fault  on  the 
middle  main.  In  large  networks  there  may  always  be  a 
slight  deflection  of  the  ammeter  pointer  when  either  P  or 
N  is  flashed,  owing  to  the  great  length,  and  consequent 
small  fault  resistance  of  the  middle  main.  The  engineer- 
in-charge  knows  approximately  the  magnitudes  of  the 
throws  to  be  expected  and  so  an  increase  in  the  value  of  the 
throw  indicates  that  a  fault  has  developed.  The  absence 
of  a  throw  also  might  indicate  that  a  fault  had  been 
cleared. 

When  the  above  operation  indicates  that  there  is  a 
fault  on  the  middle  main,  we  have  to  determine  the  portion 
of  the  main  or  feeder  on  which  the  fault  is  situated. 
If  we  put  an  ammeter  in  circuit  with  each  of  the  neutral 
feeders  in  turn,  and  notice  the  effect  in  each  case  on  the 
ammeter  pointer  of  flashing  the  outer,  a  faulty  feeder  will 
be  indicated  by  the  large  throw  obtained  when  the  testing 
ammeter  is  in  circuit  with  it.  If  all  the  feeder  mains  are 
sound,  the  fault  is  on  the  middle  distributing  main,  and  by 
putting  an  ammeter  in  circuit  at  various  points  of  the 
middle  main  in  turn,  the  first  point  at  which  no  abnormal 
throw  is  observed  on  flashing  the  outer  is  the  end  of  the 
section  of  the  middle  main  on  which  the  fault  is  situated. 

By  noticing  the  readings  of  the  ammeters  in  circuit  with 
the  feeder  mains  of  the  negative  outer,  when  the  positive 
outer  is  flashed,  a  faulty  negative  feeder  can  often  be  de- 


FAULTS   IN  NETWORKS 


149 


tected.     Similarly  a  fault  on  a  positive  feeder  is  indicated 
by  a  throw  on  its  ammeter  when  the  negative  is  flashed. 

Let  us  now  suppose  that  the  existence  of  a  fault  on  one 
of  the  negative  feeders  has  been  discovered.  The  various 
distributors  branching  from  this  feeder  should  then  be 
transferred,  one  at  a  time,  to  another  feeder.  This  may 
be  easily  done  at  the  section  pillars  or  at  the  manholes. 
The  flashing  test  is  repeated  after  each  transfer,  and  thus, 
the  faulty  distributor  is  found  when  the  transfer  stops  the 


t 

'  sa 

RA 

E 

^-J- 

^  —  *v 

© 

r\    r 

~\    r 

E 


FIG.  39.— Earth  on  the  Negative  Outer. 


throw  on  the  feeder  ammeter  due  to  flashing.  If  the  fault 
be  small,  it  may  be  necessary  to  put  a  large  resistance  in 
the  earth  connexion  of  the  middle  main  during  the  test. 
This  test  is  sometimes  laborious  owing  to  the  frequent 
journeys  to  the  station  and  back  between  each  discon- 
nexion. It  is  to  be  noticed  that  none  of  the  tests  described 
hitherto  interfere  with  the  supply  to  the  consumers. 

When  it  is  permissible  to  open  the  earth  connexion  of 


150        ELECTRIC   CABLES   AND  NETWORKS 

the  middle  main,  and  to  disconnect  various  sections  of  the 
network  in  turn,  we  may  proceed  as  follows.  Let  us  sup- 
pose, for  instance,  that  there  is  an  earth  (Fig.  39)  on  the 
negative  main.  In  this  case  there  will  be  an  appreciable 
reading  on  the  recording  ammeter  HA.  When  we  open 
the  switch  S2)  if  the  fault  be  a  bad  one,  the  potential  of 
the  positive  outer  from  earth  will  be  practically  double  of 
the  pressure  of  the  supply,  and  the  potential  of  the  middle 
main  will  be  practically  the  same  as  the  supply  pressure. 
In  this  case  a  lamp  will  burn  brightly  when  connected  be- 
tween the  middle  main  and  earth.  In  general,  if  there  is 
an  appreciable  fault  on  the  negative  main,  a  lamp  con- 
nected between  the  middle  main  and  earth  will  glow  more 
or  less  brightly.  If  then  we  connect,  at  the  nearest  net- 
work box,  a  lamp  between  the  middle  main  and  the  earth, 
the  appearance  of  the  filament  will  indicate  the  value  of 
the  voltage  V2-  The  various  service  lines  are  disconnected 
in  turn.  If  the  faulty  service  line  is  connected  with  this 
box,  there  will  be  a  sudden  change  in  the  brightness  of  the 
filament  when  it  is  disconnected.  If  not,  we  have  to  pro- 
ceed to  the  various  network  boxes  in  turn  and  repeat  the 
test,  until  the  faulty  service  line  is  discovered.  When  a 
suitable  portable  voltmeter  is  available  it  is  better  to  use 
it  instead  of  a  lamp. 

When  the  fault  is  in  the  middle  main  there  will  be  practi- 
cally no  current  indicated  by  the  ammeter  RA  in  the  earth 
connexion.  When  disconnexion  of  the  service  lines  is 
permissible  we  may  proceed  as  follows.  Open  the  switch 
S2  (Fig.  39)  and  make  a  small  artificial  leak  in  the  negative 
outer.  We  then  disconnect  the  service  lines  as  in  the  last 
section,  and  proceed  as  before,  the  only  difference  being 
that  the  lamp  will  now  glow  when  the  faulty  section  is 
disconnected. 


FAULTS   IN  NETWORKS 


151 


It  will  be  seen  that  these  methods  are  very  simple  and 
easy  to  apply.  A  drawback  to  their  use  is  the  necessity 
of  breaking  for  a  few  seconds  the  continuity  of  the  supply 
to  individual  consumers  in  the  sections  under  test. 

The  following  methods  of  locating  faults  in 
distributing  networks,  described  by  F.  Fernie, 
will  be  found  trustworthy  and  expeditious.  In  modern 
networks,  the  different  feeder  sections  are  linked  by  fuse 


General 
methods 


FIG.  40. — Arrangement  of  Switchboards  during  the  test. 

switches  in  pillars  above  ground  which  are  opened  by  a 
key.  By  sending  a  man  round  on  a  bicycle,  therefore,  it 
is  a  simple  matter  to  divide  the  network  (Fig.  40)  into  two 
distinct  sections  A  and  B,  by  removing  the  requisite  posi- 
tive, negative  and  neutral  fuses  in  the  various  pillar  boxes. 
The  neutral  fuse  is  usually  a  stout  piece  of  copper  wire. 
If  there  is  only  one  neutral  bus  bar  at  the  station,  a  second 
must  be  extemporized.  The  A  section  is  fed  by  one  set 


152        ELECTRIC   CABLES  AND   NETWORKS 

of  dynamos  and  the  B  section  by  another  set.  The  balancing 
of  the  A  section  may  be  done  by  the  storage  battery,  and 
the  balancing  of  the  B  section  by  the  balancer.  The  neutral 
bus  bars  belonging  to  the  A  and  B  sections  respectively 
are  earthed  through  separate  ammeters.  If  there  is  a 
fault  on  one  of  the  outers  the  earth  ammeter  of  the  section 
on  which  the  fault  is  situated  will  give  a  large  reading. 
Groups  of  feeders  on  this  section  together  with  the  machine 
to  which  they  are  connected  are  now  transferred  method- 
ically to  the  other.  The  earth  ammeters  are  inspected 
between  each  operation.  When  the  faulty  group  is  trans- 
ferred, the  reading  on  one  of  the  ammeters  will  suddenly 
drop  and  the  reading  on  the  other  suddenly  rise.  The 
fault  is  thus  localized  to  this  group. 

The  feeders  of  the  group  on  which  the  fault  is  situated 
are  now  transferred  back  to  the  other  section,  one  by  one, 
until  the  faulty  feeder  is  discovered.  This  faulty  feeder 
may  then  be  subdivided  further,  and  the  fault  be  localized 
to  a  few  streets.  By  disconnecting  at  the  service  boxes, 
it  can  then  be  determined  whether  it  is  in  a  consumer's 
installation  or  in  the  mains  themselves.  In  the  latter 
case,  the  fault  has  sometimes  been  detected  by  noticing  a 
dry  patch  on  a  wet  pavement. 

If  the  fault  be  on  a  neutral  feeder,  it  can  be  readily  de- 
tected by  putting  a  few  secondary  cells  or  a  booster  in 
series  with  the  earth  ammeters  in  the  A  and  B  sections 
in  turn.  If  there  be  a  fault  in  the  A  section,  it  will  be 
possible  to  get  a  large  constant  current  when  the  cells  are 
in  series  with  the  earth  ammeter  of  the  A  section.  Now 
remove  the  feeders  from  the  A  section  to  the  B  section  in 
turn.  The  faulty  feeder  will  be  detected  by  its  trans- 
ference reducing  very  considerably  the  reading  of  the  A 
ammeter. 


FAULTS   IN  NETWORKS  153 

In  this  method  a  section  of  the  network  is 
Hopkinson     isolated  and  the  readings  of  the  ammeters  on 

three  . 

ammeter      the   positive,    neutral,  and  negative   feeders   of 

this  section  are  taken.  If  the  sum  of  two  of 
the  readings  be  not  equal  to  the  third  there  must  obviously 
be  a  leak  on  the  given  section  of  the  network.  It  has  to 
be  noticed,  however,  that  even  when  the  sum  of  two  of 
them  is  approximately  equal  to  the  third  there  may  be  a 
fault  on  the  neutral  if  its  potential  is  small.  Hence  We 
must  test  for  a  neutral  leak  by  putting  a  few  cells  in  cir- 
cuit with  the  neutral  and  an  earth  connexion  in  the  manner 
described  above.  This  method  can  only  be  successfully 
used  when  the  load  is  very  steady.  If  there  is  a  motor 
load  on  the  section,  it  is  exceedingly  difficult  to  get  con- 
sistent simultaneous  readings  of  the  three  ammeters. 

If  the  system  be  a  "  drawn  in  "  system  and 
methods  of     the   fault   has   been   localized   to   a   particular 

section,  then,  if  the  cables  are  lead  covered 
and  unbraided  we  can  often  by  feeling  the  lead  at  the 
service  boxes  detect  by  the  slight  shock  generally  experi- 
enced, the  portion  of  the  cable  in  which  the  fault  is  situated. 
Sometimes,  also,  more  especially  with  rubber  and  vulcanized 
bitumen  cables,  the  fault  can  be  localized  by  the  smell  of 
the  overheated  insulating  material.  If,  however,  the 
cables  are  "  solid-laid,"  or  are  armoured  and  laid  direct 
in  the  ground,  one  or  other  of  the  following  methods,which 
have  been  found  useful  in  practical  work,  can  be  used. 

Let  us  suppose  that  the  fault  lies  in  the  parti- 
The  fall  of 
potential       cular  loop  LFM  (Fig.  41).     Earth  one  pole  of 

method 

the  battery  and  connect  the  other  through  a 
resistance  R,  and  an  ammeter  A ,  with  the  end  L  of  the  main 
LM .  As  ERLFE  forms  a  closed  circuit,  a  current  will 
flow  which  can  be  read  on  the  ammeter.  An  electros tatiQ 


154        ELECTRIC   CABLES   AND  NETWORKS 

voltmeter  placed  between  L  and  M  will  read  the  voltage 
V  between  L  and  F,  for  F  and  M  are  at  the  same  potential 
since  there  is  no  current  in  FM .  If  x  be  the  resistance  of 
LF,  we  have  x  =  V/At  where  A  is  the  ammeter  reading, 


E 

FIG.  41. — Fall  of  Potential  Method. 

and  thus  x  is  found.  Knowing  the  resistance  of  a  yard  of 
cable  the  distance  to  the  fault  can  now  be  readily  calcu- 
lated with  considerable  accuracy. 

It  is  to  be  noticed  that  we  have  made  the  assumption 
that  there  is  only  one  fault  in  the  main.  If  there  were 
more  than  one  fault,  this  calculated  length  would  be  greater 
than  the  distance  to  the  first  fault.  It  is  therefore  always 
advisable  to  repeat  the  test,  connecting  the  end  M  of  the 
cable  with  the  battery.  If  the  two  results  obtained  on 
the  assumption  that  there  is  a  single  fault  agree  in  locating 
this  fault  at  the  same  point  of  the  cable,  it  is  highly  pro- 
bable that  a  fault  will  be  found  at  this  point. 

When  a  spare  drum  of  the  same  cable  as  the  faulty  main 
is  available  at  the  station,  the  following  modification  of  the 
above  test  will  be  found  convenient.  Replace  the  ammeter 
A  and  the  resistance  E  (Fig.  41)  by  the  drum  of  cable,  every- 
thing else  remaining  as  before.  Take  the  reading  V  of 
the  voltmeter  when  placed  across  the  terminals  of  the  drum 
and  the  reading  V  between  L  and  M .  Then  if  I'  be  the 


FAULTS   IN  NETWORKS 


155 


length,  of  the  cable  on  the  drum  and  I  the  distance  LF  to 
the  fault  we  have  l=(V/V')l'. 

The  principle  of  the  loop  test  can  be  readily 
understood  from  Fig.  42.  A  wire  bridge  pq 
is  connected  across  the  terminals  L  and  M  of  the  loop  of 
cable  in  which  the  fault  lies,  and  a  galvanometer  G  is  also 
placed  between  the  terminals.  One  pole  of  a  battery  is 
connected  with  the  jockey  of  the  bridge,  the  other  pole 
being  earthed.  The  jockey  is  then  moved  about  until  the 
deflection  of  the  galvanometer  is  zero.  In  this  case  we 


1 


^  E 


FIG.  42.—  Bridge  Method  of  Testing. 

have    x/y=p/q,   and    thus    y  =  {q/(p-\-q)}(x-\-y).     Hence, 
if  I  be  the  length  of  the  whole  loop,  the  length  of  LF  is 


In  this  test  care  has  to  be  taken  that  the  resistance  of 
the  connexions  at  L  and  M  are  negligibly  small.  It  is 
immaterial  whether  the  fault  at  F  is  polarizable  or  not  as 
it  is  in  series  with  the  battery.  This  is  one  of  the  most 
generally  useful  and  accurate  of  the  methods  of  localizing 
faults. 

In  certain   cases   the  following   test   can   be 
easily    applied.     Place    ammeters   Al  and    A2 
(Fig.  43),  of  negligible  resistance,  in  series  with 
each  branch  of  the  loop.     Now  connect  L  with  one  pole 


Two 

ammeter 
method 


156        ELECTRIC  CABLES   AND  NETWORKS 

of  a  battery,  the  other  pole  of  which  is  earthed.  If  A^ 
and  A 2  be  the  readings  of  the  ammeters,  then,  by  Ohm's 
law,  the  potential  difference  between  L  and  F  will  equal 
xAi,  and  it  will  also  equal  yA2.  Hence  y(Al-\-A2)  =(x-{-y)Ai, 
and  thus,  if  I  be  the  total  length  of  the  loop,  the  distance 
LF  to  the  fault  will  equal  { 


FIG.  43. — Two  Ammeter  Method. 


In  the  above  tests,  it  has  been  assumed  that  the  section 
of  the  cable  is  uniform  throughout.  If  it  is  not  uniform, 
they  can  still  be  applied,  provided  the  lengths  and  the 
resistances  of  the  various  portions  of  the  cables  are  known. 
Let  us  suppose  that  the  lengths  LL±,  L±L2,  . . .  are  of 
different  sections  and  that  their  resistances  are  Rlt  B2,  . . . 
respectively.  Let  us  also  suppose  that  the  resistance  y 
of  LF  lies  in  value  between  R^-\-E2  and  B1-\-R2-\-R3.  The 
fault  will  obviously  be  on  the  section  L2L3,  and  the  resist- 
ance of  the  conductor  between  L2  and  the  fault  will  be 
y — (Ri+Rz).  Hence,  knowing  the  resistance  per  yard 
of  the  section  L2L3  of  the  cable,  we  can  easily  find  the 
position  of  the  fault. 

induction          The   following    "  induction   method "   is   ex- 
method        tensively  used  in  practice.     The  great  advan- 
tage of  this  method  is  that  no  knowledge  is  required  of 


FAULTS  IN  NETWORKS 


157 


the  resistance  of  the  main  under  test,  and  so  uncertainties 
due  to  partial  breaks  or  bad  joints  do  not  affect  it.  It 
can  also  be  applied  when  there  are  several  earth  faults  on 
the  cable. 

Let  us  suppose  that  there  is  an  earth  fault  at  F  in  the 
cable  LM  (Fig.  44).     Insulate  one  end  M  of  the  cable  and 


FIG.  44. — Induction  Method. 

connect  the  other  end  with  the  terminal  of  a  generator  of 
alternating  or  pulsating  E.M.F.  If  the  other  end  of  this 
generator  be  connected  with  the  earth,  an  interrupted 
current  will  flow  in  the  cable  to  the  fault  and  return  to 
the  generator  by  the  earth.  C  is  a  wooden  triangular 
framework  wound  with  several  turns  of  insulated  wire 
the  ends  of  which  are  connected  with  a  telephone  T.  When 
the  plane  of  the  triangle  is  vertical  and  its  base  is  parallel 
to  the  cable  carrying  the  interrupted  current  the  continual 
fluctuation  of  the  lines  of  magnetic  induction,  linked  with, 
the  triangular  coil,  will  induce  a  fluctuating  E.M.F.  in  it, 
and  so  a  buzzing  sound  will  be  heard  in  the  telephone  re- 
ceiver. If  the  framework,  held  in  this  manner,  be  carried 
or  wheeled  directly  over  and  along  the  route  of  the  cable, 
a  cessation  or  change  of  the  note  generally  indicates  the 
position  of  the  fault.  If  the  conductor  carrying  the  current 


158        ELECTRIC   CABLES   AND  NETWORKS 

be  enclosed  in  a  lead  sheath  or  in  metallic  casing,  the 
earthed  terminal  of  the  generator  should  be  connected 
with  the  sheath  or  the  casing. 

Biavier's  When  the  resistance  of  the  line  is  high,  the 
following  method  is  sometimes  useful.  Let 
us  suppose  (Fig.  45)  that  there  is  a  fault  of  resistance  z  at 
F.  Let  the  resistances  also  of  LF  and  FM  be  x  and  y 
respectively.  In  practice,  the  resistance  R  of  the  whole 
line  LM  is  known. 

WVWV — TM 


E 

FIG.  45. — Biavier's  Method. 

We  first  measure  at  L  the  insulation  resistance  R±  of 
the  line,  when  the  end  M  is  insulated.  We  next  measure 
its  insulation  resistance  E2,  when  the  end  M  is  connected 
directly  with  the  earth.  We  thus  obtain  the  three  following 
equations  to  find  x,  y  and  z. 

x+y=R  (1), 

x+z=E,          (2), 

and  x+yz/(y+z)=E2          (3). 

Eliminating  y  and  z  from  (3),  by  means  of  (1)  and  (2), 

we  have 

x+(E—x)(Ei—x)/(E+Ei—2x)  =R2, 

and  thus,  by  solving  this  quadratic  equation  we  get 

x  =E2—  { (E—E2)(E,—E2) }  V 2, 

the  negative  sign  being  prefixed  to  the  radical,  since  by 
the  conditions  of  the  problem  yz/(y+z)  can  only  be  posi- 
tive, and  hence,  by  (3),  x  must  be  less  than  E2. 


FAULTS   IN   NETWORKS  159 

There  are  four  assumptions  made  in  this  test  that  have 
to  be  remembered  in  practical  work.  In  the  first  place 
we  assume  that  there  is  only  one  fault,  secondly  that  the 
fault  is  non  polarizable,  thirdly  that  the  temperature  of 
the  mains  is  uniform,  and  fourthly  that  the  resistances 
are  so  high  that  the  errors  due  to  imperfect  contacts 
are  negligibly  small.  In  electric  lighting  networks  the 
fourth  assumptipn  is  the  most  serious  and  in  many  cases  it 
is  not  permissible. 

In  a  power  transmission  line,  R  was  44  ohms, 
Example 

and  the  values  of  Et  and  E2  found  by  measure- 
ment were  25-9  and  24-3  respectively.     In  this  case 

x  =24-3—  { (25-9— 24-3)( 44—24-3) } J/ 2 

=  18-7. 

By  (2),  18-7+z=25-9, 

and  therefore,  2  =  7*2. 

Similarly,  by  (1),       y=25-3. 

When  tests  can  be  made  at  both  ends  of  the  line  it  is 
advisable  to  make  them.  If  the  results  differ  largely  it 
is  probable  that  there  is  more  than  one  fault. 

REFERENCES. 

F.  Charles  Raphael.         The  Localization  of  Faults  on  Electric  Light 

Mains. 

J.  A.  Fleming.      Handbook  for  the  Electrical  Laboratory  and  Testing 
Room. 

G.  L.  Black.     "  The  Maintenance  of  Underground  Mains."     Electri- 

cian, vol.  56,  p.  507,  1906. 
F.   Fernie.     "  Localization  of  Faults  on  a  Three-Wire  Network." 

Electrical  Review,  vol.  60,  p.  451,  1907. 
A.  Schwartz.     "'  Flexibles,'  with  Notes  on  the  Testing  of  Rubber." 

J.I.E.E.,  vol.  39,  p.  31,  1907. 


DIELECTRIC  STRENGTH 


M 


CHAPTER    VIII 

Dielectric   Strength 

Dielectric  strength — Disruptive  discharge — Spherical  condenser — 
Single  core  main — Effect  of  shape  of  conductor — Equipotential 
surfaces  round  a  pointed  conductor — Spherical  electrodes — 
Composite  dielectrics — The  maximum  electric  stress  between 
equal  spherical  electrodes — American  rules — Failing  cases  in 
practice — Measuring  the  dielectric  strength  of  gases — Dielectric 
strength  of  liquids — Dielectric  strength  of  isotropic  solids — 
Dielectric  strength  of  eolotropic  solids — References. 

Dielectric  ^N  power  transmission,  whether  by  direct  or 
strength  alternating  current,  the  saving  of  copper  effected 
by  using  very  high  pressures  has  directed  the  attention 
of  manufacturers  to  the  construction  of  cables  which  will 
withstand  pressures  of  100  kilovolts  and  upwards.  To 
design  cables  which  will  successfully  withstand  these  pres- 
sures, a  knowledge  of  the  electric  stresses  to  which  the 
various  insulating  materials  round  the  core  will  be  sub- 
jected under  working  conditions  is  essential,  as  well  as  an 
accurate  knowledge  of  the  dielectric  coefficient  (specific 
inductive  capacity),  dielectric  strength,  and  resistivity  of 
each  of  the  insulating  wrappings. 

In  this  chapter  we  shall  discuss  the  dielectric  strength 
of  insulating  materials  and  the  methods  of  measuring  it. 
By  the  dielectric  strength  of  an  isotropic  insulating  material 
in  a  given  physical  condition  is  meant  the  maximum  value 
of  the  electric  stress  which  it  can  withstand  without  break- 

163 


L64        ELECTRIC  CABLES   AND  NETWORKS 

ing  down.  The  substance  of  a  homogeneous  solid  is  called 
isotropic  (see  p.  3)  when  a  spherical  portion  of  it  tested  by 
any  physical  agency  exhibits  no  difference  in  quality  how- 
ever it  is  turned.  Substances  which  are  not  isotropic  are 
called  eolo tropic.  From  an  electrical  point  of  view  we  can 
regard  gases  and  pure  liquids  as  isotropic. 

In  his  Experimental  Researches  in  Electricity,  vol.  i,  p. 
436,  Faraday  states  that  "  discharge  probably  occurs  not 
when  all  the  particles  have  attained  to  a  certain  degree 
of  tension,  but  when  that  particle  which  has  been  most 
affected  has  been  exalted  to  the  subverting  or  turning 
point."  He  therefore  considers  that  there  is  a  definite 
"  subverting  or  turning  point  "  for  each  particle  of  the 
material ;  that  is,  that  it  has  a  definite  dielectric  strength. 

In  order  to  test  Faraday's  conclusion  it  is  necessary  to 
be  able  to  calculate  the  electric  stress  at  the  point  be- 
tween two  electrodes  where  it  has  its  maxiumm  value. 
We  shall  first  consider  how  electric  stress  is  measured. 

From  symmetry,  it  is  obvious  that  the  equipotential 
surfaces  round  a  charged  spherical  conductor  surrounded 
by  a  homogeneous  dielectric,  and  at  a  considerable  distance 
from  all  other  conductors,  are  spherical  in  shape.  'If  q 
denote  the  charge  on  the  conductor,  and  v  the  potential 
at  a  point  at  a  distance  r  from  the  centre  of  the  sphere, 
we  have  v=q/r,  and  therefore  the  potentials  of  the  equi- 
potential surfaces  surrounding  the  sphere  vary  inversely 
as  their  radii.  Let  us  suppose,  for  example,  that  the 
spherical  conductor  shown  in  Fig.  46  is  at  a  potential  of 
10,000  volts,  then  the  potentials  of  the  various  circles 
drawn  in  the  figure  are  9,000,  8,000,  . . .  1,000,  volts 
respectively.  The  equipotential  surface  of  zero  potential 
would  be  at  infinity.  It  is  to  be  noticed  that  close  up 
to  the  sphere  the  surfaces  are  crowded  together.  The 


DIELECTRIC  STRENGTH 


165 


spherical  stratum  of  the  dielectric  included  between  the 
conductor  and  the  first  equipotential  surface  is  obviously 
subjected  to  the  greatest  stress.  The  average  stress  on 
any  of  the  spherical  layers  shown  in  Fig.  46  is  inversely 
proportional  to  its  thickness.  By  increasing  the  number 
of  equipotential  surfaces  indefinitely  until  the  concentric 


FIG.  40. — Section  of  the  equipotential  surfaces  round  a  charged  sphere, 
the  potential  difference  between  consecutive  surfaces  being  constant. 

layers  of  the  dielectric  become  indefinitely  thin,  we  see 
that  the  electric  stress  at  a  point  is  measured  by 
{v — (v+dv)} /dr,  that  is,  by  — dv/dr.  This  quantity  is 
called  by  electricians  the  potential  gradient  at  the  point. 

It  follows  at  once  from  the  definition  of  potential  that 
the  resultant  force  at  a  point  in  a  dielectric  is  equal  to 
the  rate  at  which  the  potential  diminishes  as  we  move 


166        ELECTRIC  CABLES  AND  NETWORKS 

along  the  line  of  force  through  the  point.  Hence  the 
potential  gradient  is  the  resultant  force,  and  is  the  force 
with  which  a  unit  positive  charge  placed  at  the  point  would 
be  urged  if  it  could  be  placed  there  without  disturbing 
the  distribution  elsewhere.  This  force  measures  the  electric 
stress  on  the  dielectric. 

A  good  way  of  picturing  what  happens  in  a  dielectric  is 
by  means  of  Faraday's  tubes  of  force.  We  picture  one 
end  of  one  of  these  tubes  anchored  to  a  unit  positive  charge 
on  the  positive  electrode,  and  the  other  end  anchored  to  a 
unit  negative  charge  on  the  other  electrode. 

In  the  case  of  the  spherical  conductor  considered  above, 
q  tubes  will  start  from  the  surface  of  the  sphere,  and  thus 
the  number  of  tubes  passing  through  a  square  centimetre 
of  the  equipotential  surface  which  is  at  a  distance  r  from 
the  centre  of  the  sphere  is  q/^jrr2.  But  the  potential 
gradient  at  a  distance  r  from  the  centre  of  the  sphere  is 
— q/r2,  and  hence  4?r  times  the  number  of  Faraday  tubes 
which  pass  through  unit  area  of  the  equipotential  surface 
is  the  numerical  value  of  the  potential  gradient  at  all  points 
on  that  surface.  Another  name  for  the  electric  stress  at  a 
point  is  the  electric  "  intensity  "  at  the  point.  It  has  to  be 
remembered  when  reading  the  literature  of  the  subject  that, 
"  the  resultant  electric  force,"  "  the  potential  gradient "  and 
"  the  electric  intensity  "  are  all  used  to  denote  the  re- 
sultant electric  stress  R  at  a  point.  In  symbols, 

E=  —  dv/dr=  ±7rN. 

If  R  were  constant  over  the  equipotential  surface  passing 
through  the  point  under  consideration,  N  would  be  the  num- 
ber of  Faraday  tubes  per  square  centimetre  of  this  surface. 
Disruptive          Greatly  to  the  disappointment  of  the  earlier 

discharge  physicists,  it  Was  found  that  the  disruptive 
voltage  between  metal  electrodes  when  close  together 


DIELECTRIC   STRENGTH  167 

was  apparently  not  governed  by  the  maximum  value  of 
the  potential  gradient  between  them.  As  early  as  1860, 
however,  Lord  Kelvin  was  led  to  infer,  from  his  experi- 
ments with  pressures  of  between  5  and  6  kilovolts  obtained 
from  a  battery  of  5510  Daniell  cells  in  series,  that  at  high 
voltages  the  disruptive  discharge  between  large  metal 
electrodes  will  take  place  the  moment  the  electric  stress 
attains  a  definite  value.  Recent  experiments  at  high 
voltages  have  amply  confirmed  Lord  Kelvin's  conclusion, 
and  it  forms  the  basis  of  the  practical  methods  of  measuring 
dielectric  strengths. 

It  has  to  be  remembered  that  part  of  an  insulating 
material  can  break  down  without  a  disruptive  discharge 
necessarily  ensuing.  When  brush  discharges,  for  instance, 
occur  from  an  electrode  in  air  part  of  the  air  surrounding 
the  electrode  has  become  a  true  gaseous  electrolyte,  and 
its  insulativity  therefore  has  broken  down.  The  air  at  the 
boundary  of  this  electrolyte  has  not  broken  down,  because 
the  electric  stress  to  which  it  has  been  subjected  has  not 
reached  the  "  subverting "  value,  which  measures  the 
dielectric  strength  of  the  medium. 

In  Fig.  46,  when  the  stress  close  up  to  the  sphere  equals 
the  dielectric  strength  of  air,  which  is  about  38  kilovolts 
per  centimetre  at  ordinary  temperatures  and  pressures, 
the  spherical  layer  round  it  breaks  down  and  becomes 
luminous.  If  we  raise  the  potential  still  higher  the  sphere 
is  surrounded  by  a  luminous  spherical  envelope  called  the 
corona,  the  radius  of  which  is  proportional  to  the  potential 
to  which  we  raise  the  conductor. 

Spherical          *n  ^e  case  °^  a  spherical  condenser  we  have 

condenser  a  metallic  sphere  concentric  with  a  metallic 
spherical  envelope.  If  the  radius  of  the  inner  sphere  be 
a,  and  the  radius  of  the  outer  sphere  be  b,  we  have  v=q/r, 


168        ELECTRIC  CABLES  AND  NETWORKS 

and  q  =  Vab/(b  —  a),  where  F  is  the  potential  difference 
between  the  two  conductors.  The  equipotential  surfaces 
are  thus  the  same  as  in  Fig.  46,  and  since 

R  =—dv/dr  =  Vab/r*(b—a), 

we  see  that  R  has  its  maximum  value  Rm,  when  r=a. 
Hence  Km  =  Vb/a(b—a). 

If  we  suppose  that  the  sphere  is  surrounded  by  air,  then, 
when  Rm  attains  the  value  of  the  dielectric  strength  of 
air  Rmax^  the  air  surrounding  the  sphere  breaks  down  and 
becomes  a  conductor,  but  a  disruptive  discharge  does  not 
necessarily  ensue.  If  the  breaking  down  of  the  first 
stratum  of  air  makes  the  new  value  of  Rm  equal  to  or 
greater  than  jRmax<,  a  disruptive  discharge  will  ensue  but 
if  it  makes  it  less  than  Rmaaif  there  will  be  no  disruptive 
discharge,  and  a  corona  will  be  formed. 

If  «i  be  the  radius  of  the  corona  formed,  we  have 


and  thus  dV/da^  =  (b—2a1)RmaXf/b, 

assuming  that  Rmax.  and  b  are  constant.  We  see  that  V 
increases  with  «i  until  a^  gets  equal  to  6/2,  it  then  diminishes 
as  «i  increases.  Hence  a  corona  can  exist  if  «i  be  less 
than  6/2,  for  the  value  of  Rm  in  the  stratum  immediately 
surrounding  the  corona  is  less  than  RmaXm  It  cannot,  how- 
ever, exist  if  «!  be  greater  than  6/2,  for  the  value  of  Rm  in 
the  stratum  surrounding  it  would  be  greater  than  RUM,.. 
We  see,  therefore,  that  the  size  of  the  inner  sphere  has  no 
practical  effect  on  the  disruptive  voltage  provided  that 
its  radius  be  less  than  6/2.  We  see  also  that  a  spherical 
condenser  can  be  used  to  measure  the  dielectric  strengths 
of  gases  or  liquids  provided  that  the  radius  of  the  inner 
conductor  be  not  less  than  6/2.  In  this  case 


DIELECTRIC  STRENGTH 


169 


where    V  is   the  voltage   between  the   conductors  at  the 
instant  of  the  discharge. 

Single  core         *n  ^%-  ^  tne   equipotential  surfaces  for   a 

mam        single  core  cable  with  a  homogeneous  dielectric 

are  shown.     Let  us  suppose  that  a  is  the  radius  of  the 


FIG.  47. — Section  of  the  equipotential  surfaces  in  a  single  core  cable- 
The  dotted  circle  is  the  outer  radius  of  the  broken-down  dielectric 
at  the  instant  of  the  disruptive  discharge. 

cylindrical  core,  and  that  b  is  the  inner  radius  of  the  coaxial 
cylindrical  lead  sheath.  The  potential  gradient  R  at  a 
point  P  in  the  dielectric  which  is  distant  r  from  the  axis 
of  the  core,  is  given  by 

E  =—(  1  /X)dv/dr  =  F/rloge(6/a), 
where  V  is  the  potential  difference  between  the  core  and 


170        ELECTRIC  CABLES  AND  NETWORKS 

the  sheath,  and  \  is  the  dielectric  coefficient  of  the  insu- 
lating material.  R  has  its  maximum  value  Rm  when 
r=  a,  and  thus  Rm  =  V/aloge(b/a). 

Let  us  first  suppose  that  the  insulating  material  is  a 
gas  of  dielectric  strength  EmaXm.  The  conditions  that  a 
cylindrical  corona  of  radius  a±  be  formed  are  that 


and  that  dv/da  is  a  positive  quantity  when  a  =a±.  The 
second  condition  is  true  when  loge  (6/tfi)  —  1  is  positive, 
that  is,  when  «i  is  less  than  6/e,  where  e  =2*718  ...  is 
the  base  of  Neperian  logarithms. 

When  the  insulating  material  is  a  homogeneous  solid  sub- 
stance of  dielectric  strength  R^^  the  same  formulae  apply, 
at  least  to  a  first  approximation.  If  the  radius  of  the 
core  be  less  than  b/e,  then,  when  F  is  greater  than 
a  loge(b/a)Rmax,  and  less  than  (b/e)Rmax.,  some  of  the 
dielectric  surrounding  the  inner  core,  which  we  suppose 
to  be  a  smooth  cylinder,  has  broken  down  and  become  a 
conductor.  When  the  voltage  V  exceeds  (b/e)Rma,K  a 
disruptive  discharge  will  ensue. 

A  comparison  of  Figs.  46  and  47  will  show  that  the 
electric  stresses  close  to  a  spherical  conductor  are  greater 
than  close  to  a  cylindrical  conductor  of  the  same  radius 
and  at  the  same  potential.  It  will  be  noticed  that  the 
equipotential  surfaces  are  more  crowded  together  round 
the  sphere  than  round  the  cylinder.  Since  we  have  supposed 
the  cylinder  to  be  infinitely  long,  the  change  of  potential 
as  we  recede  from  it  will  obviously  not  be  so  rapid  as  in 
the  case  of  the  finite  sphere. 

In  Fig.  48  the  effect  of  the  shape  of  a  con- 
Effect  of 

shape  of      ductor  on  the  electric  stresses  in  the  medium 
conductor 

surrounding    it    is    illustrated.     In    the    ngure 

the    potential    difference    between    any    adjacent   pair    of 


DIELECTRIC   STRENGTH 


171 


FIG.  48. — Section  of  the  equipotential  surfaces  when  the  core  is  a  strip 
of  copper  with  rounded  ends. 


\ 


\ 


5,000 


2.600 


FIG.   49. — Section  of  the  equipotential  surfaces  round  a  tapered  copper 
conductor  maintained  at  30  kilovolts. 


172        ELECTRIC  CABLES   AND  NETWORKS 

equipotential  surfaces  is  the  same.  The  section  of  the 
core  is  elliptical  in  shape,  and  the  maximum  value  of  the 
electric  stress  at  the  rounded  ends  of  the  core  is  ten  times 
the  minimum  stress  at  the  middle  part  of  the  core.  The 
equipotential  surfaces  show,  however,  that  the  electric 
stress  is  practically  constant  in  the  layer  of  the  insulating 
material  next  the  lead  sheath. 

In  Fig.  49,  the  equipotential  surfaces  round 

Equipoten- 

tial  surfaces    a    tapered     copper    conductor,     ellipsoidal     in 

pointed       shape,  are   shown.     The  electric   stress  on  the 

insulating  material  in  contact  with  the  rounded 

point  is  very  great.     When  electrodes  of  this  shape  are 

used    for    testing    it    is    extremely    difficult    to    calculate 


FIG.  50.  —  Section  of  the  equipotential  surfaces  round  two  spheres  having 
equal  and  opposite  charges.  The  potential  gradient  in  the  dielectric 
is  obviously  greatest  at  the  points  of  the  electrodes  which  are  closest 
together. 

the  value  of  the  maximum  potential  gradient,  and  hence, 
only  rough  comparative  tests  can  be  made  with  them. 
Spherical          *n  ^§-  50'  ^e   equipotential   surfaces  round 


electrodes      ^wo   spherical   electrodes    maintained   at   equal 
and  opposite  potentials  are  shown.     The  potential  difference 


DIELECTRIC  STRENGTH 


173 


between  any  two  adjacent  surfaces  is  the  same.  The 
potential  gradient  is  obviously  a  maximum  at  the  points 
of  the  electrodes  which  are  closest  together.  The  value 
of  the  potential  gradient  at  these  points  can  be  easily  cal- 
culated by  means  of  the  tables  given  below.  It  can  be 
shown  that  if  the  spheres  be  not  farther  apart  than  twice 
their  diameter,  a  disruptive  discharge  will  take  place  the 
moment  the  portions  of  the  insulating  material  which  are 


FIG.     51. — Flux    lines    between    cylindrical    electrodes.     The    potential 
gradient  is  a  maximum  at  the  corners  of  the  electrodes. 


subjected  to  the  maximum  stress  break  down.  Hence, 
the  disruptive  voltage  enables  us  to  find  the  dielectric 
strength  of  the  medium  by  which  they  are  surrounded. 
For  practical  testing,  spherical  electrodes  are  generally 
the  best. 

Composite         ^ke  effect  of  introducing  a  piece  of  insulating 
dielectrics     material  between  two   metal  electrodes   main- 
tained at  constant  potentials  is  illustrated  in  Figs.  51  and 


174        ELECTRIC  CABLES   AND  NETWORKS 

52.  The  insulating  material  is  supposed  to  have  a  high 
dielectric  coefficient.  The  capacity  and  consequently  the 
number  of  Faraday  tubes  between  the  electrodes  is  con- 
siderably increased.  The  stress  on  the  air  which  is  measured 
by  4?r  times  the  number  of  Faraday  tubes  per  unit  area  is 
increased  also.  Hence  the  introduction  of  a  piece  of  glass 
near  the  electrodes  sometimes  causes  a  disruptive  discharge 


FIG.  52. — Flux  lines  when  a  glass  sphere  is  introduced  between  the  elec- 
trodes, the  potential  difference  being  maintained  the  same  as  in 
Fig.  51.  Notice  the  increase  in  the  total  flux,  and  consequently 
the  increase  in  the  stress  on  the  dielectric. 


between  them.  Since  the  dielectric  coefficient  of  a  metal 
is  infinite,  introducing  a  metal  conductor  between  the 
electrodes  increases  the  stress  more  than  a  piece  of  insu- 
lating material  of  the  same  size  would.  The  calculation 
of  the  maximum  potential  gradient  when  the  insulating 
materials  have  different  dielectric  coefficients  is  in  general 
very  difficult. 


DIELECTRIC   STRENGTH  175 

The    easiest    way    of    finding    the    dielectric 

maximum     strength  of  insulating  materials  is  by  finding 

6  stres?        the     disruptive     voltage     between    two     equal 


spherical  electrodes  embedded  in  the  material. 
The  author  has  shown  (Phil.  Mag.  (6),  vol.  ii, 
p.  258,  1906),  that,  if  the  spheres  be  at  a  less 
distance  apart  than  twice  the  diameter  of  either,  a  disruptive 
discharge  will  ensue  the  moment  the  maximum  electric 
stress  between  the  spheres  equals  the  dielectric  strength 
of  the  material.  In  order  to  calculate  the  maximum 
electric  stress  at  the  instant  of  discharge  we  must  know 
the  potential  and  size  of  each  sphere  and  the  distance 
between  them. 

Let  a  be  the  radius  of  each  sphere,  and  let  x  be  the  mini- 
mum distance  between  them.  Let  us  first  suppose  that 
one  sphere  is  at  the  potential  FI  and  that  the  other  is  at 
zero  potential.  In  this  case  the  maximum  electric  stress, 
It  max.,  between  them  is  given  by 

JWp.=(Fi/*)/i. 

where  the  values  of  /i  can  be  found  from  Table  II.     A 

proof  of  this  formula  is  given  in  the  author's  paper  (quoted 
above).  In  the  important  practical  case  when  Fi  =  —  F2 
=  F/2,  where  F2  is  the  potential  of  the  second  sphere,  we 
have 


where  /  can  be  found  from  Table  I. 

In  general,  if  Fi  and  F2  be  the  potentials  of  the  two 
spherical  electrodes,  and  Fi  be  numerically  greater  than 
F2,  we  have 


where  /  and  /i  are  functions  of  x/a,  the  values  of  which  can 
be  found  from  Tables  I  and  II. 

Hence  by  this  formula  we  can  calculate  the  dielectric 


176       ELECTRIC  CABLES  AND  NETWORKS 


strength  of  the  material  from  the  potentials  of  the  electrodes 
at  the  instant  of  the  disruptive  discharge. 

TABLE  I. 
VALUES  OF  /. 


x/a. 

/. 

x/a. 

/. 

o-o 

•000 

2 

1-770 

O'l 

•034 

3 

2-214 

0-2 

•068 

4 

2-677 

0-3 

•102 

5 

3-151 

0-4 

•137 

6 

3-632 

0-5 

•173 

7 

4-117 

0'6 

•208 

•    8 

4-604 

0-7 

•245 

9 

5-095 

0-8 

•283 

10 

5-586 

0-9 

•321 

100 

50-51 

1-0 

•359 

1,000 

500-5 

1-5 

•559 

10,000 

5,000-5 

TABLE  II. 

VALUES  OF 


x/a. 

A- 

x/a. 

/I. 

o-o 

•000 

2 

2-339 

o-i 

•034 

3 

3-252 

0-2 

•068 

4 

4-201 

0-3 

•106 

5 

5-167 

0-4 

•150 

6 

6-143 

0-5 

1-199 

7 

7-125 

0-6 

1-253 

8 

8-111 

0-7 

1-313 

9 

9-100 

0-8 

1-378 

10 

10-091 

0'9 

1-446 

100 

100-0 

1-0 

1-517 

1,000 

1,000 

1-5 

1-909 

10,000 

10,000 

DIELECTRIC   STRENGTH 


177 


The 

strengthof 
air 


In  the  following  table  the  values  of  x  and 

V  are  taken  from  Dr-  Zenneck's  work,  Elek- 
tromagnetische  ScJiwingungen  und  Drahtlose 
Telegraphic,  1905,  p.  1011.  They  are  due  to  J.  Algermissen 
and  are  deduced  from  the  average  of  the  values  obtained 
on  different  days  under  varying  conditions.  It  has  been 
assumed  that  the  potentials  of  the  electrodes  are  +F/2 
and  —  F/2  respectively  at  the  instant  of  discharge.  As 
the  results  in  the  last  column  are  very  approximately  con- 
stant the  assumption  is  justified  :  — 


TABLE  III. 

J.  Algermissen.     5-cm.  spheres  (a  =2-5).     x  is  measured 
in  centimetres,  and  V  in  kilo  volts. 


X. 

x/a. 

/(calc.). 

V(obs.). 

Rmax  (calc.). 

2-0 

0-80 

1-283 

58-2 

37-3 

2-2 

0-88 

•312 

62-8 

37-5 

2-4 

0-96 

•342 

67-0 

37-5 

2-6 

1-04 

•374 

70-8 

37-4 

2-8 

1-12 

•406 

74-4 

37-4 

3-0 

1-20 

•437 

78-0 

37'4 

3-2 

1-28 

•469 

81-3 

37-3 

3-4 

1-36 

•500 

84-7 

37-4 

3-6 

1-44 

•533 

88-0 

37'5 

3-8 

1-52 

•566 

91-2 

37-6 

4-0 

1-60 

•599 

94-2 

37-7 

4-2 

1-68 

•632 

97-2 

37-5 

has  been  calculated  by  means 


In  the  above  table 
of  the  formula 


From  the  above  results,  and  from  many  other  experi- 
mental results  obtained  with  both  alternating  and  direct 
pressures,  the  author  concludes  that  the  dielectric  strength 

N 


178        ELECTKTC  CABLES  AND  NETWORKS 

of  air  under  normal  conditions  is  about  3-8  kilo  volts  per 
millimetre. 

American  ^ke  practical  constancy  of  the  dielectric  strength 
rules  of  ajr  un(jer  ordinary  atmospheric  conditions  is 
recognized  in  the  Standardization  Rules  (1907)  of  the 
A.I.E.E.  For  instance,  in  §  243,  when  discussing  the  value 
of  the  spark-gap  safety-valve,  it  is  stated  that  "  a  given 
setting  of  the  spark-gap  is  a  measure  of  one  definite  voltage, 
and,  as  its  operation  depends  upon  the  maximum  value  of 
the  voltage  wave,  it  is  independent  of  wave  form,  and  is  a 
limit  on  the  maximum  stress  to  which  the  insulation  is 
subjected.  The  spark-gap  is  not  conveniently  adapted 
for  comparatively  low  voltages."  The  reason  for  the 
limitation  given  in  the  last  sentence  of  the  above  quotation 
will  be  discussed  below. 

In  Appendix  D  of  the  American  Rules,  the  following 
table  of  the  sparking  distances  in  air  between  "  opposed 
sharp  needle-points "  for  sine-shaped  voltage  waves  is 
given.  The  numbers  are  calculated  from  the  experimental 
results  given  in  a  paper  on  the  "  Dielectric  Strength  of 
Air,"  by  Professor  Steinmetz,  published  in  the  Transactions 
of  the  American  Institute  of  Electrical  Engineers,  vol.  xv, 
p.  281.  Under  normal  atmospheric  conditions,  with  new 
sewing  needles  supported  axially  at  the  end  of  linear  con- 
ductors which  are  at  least  twice  the  length  of  the  gap,  the 
maximum  difference  between  the  observed  disruptive 
voltages  and  the  values  given  in  the  table  will  probably 
be  well  within  5  per  cent,  of  either  voltage.  Care  must 
be  taken  that  the  potentials  of  the  needles  are  always  equal 
and  opposite  and  that  no  foreign  body  is  near  the  spark- 
gap.  In  practical  work  it  is  also  important  that  a  non- 
inductive  resistance  of  about  one-half  of  an  ohm  per  volt 
should  be  inserted  in  series  with  each  terminal  of  the  gap 


DIELECTRIC   STRENGTH 


179 


so  as  to  keep  the  discharge  current  between  the  limits  of 
one-quarter  ampere  and  two  amperes.  The  object  of  limit- 
ing this  current  is  to  prevent  the  surges  of  the  voltage 
and  current  which  might  otherwise  occur  at  the  instant  of 
breakdown. 

TABLE  IV. 

SPARKING    DISTANCES    BETWEEN    NEEDLE   POINTS. 


Effective 
Kilovolts. 

Inches. 

Cms. 

Effective 
Kilovolts. 

Inches. 

Cms. 

5 

0-225 

0-57 

140 

13-95 

35-4 

10 

0-47 

1-19 

150 

15-0 

38-1 

15 

0-725 

1-84 

160 

16-05 

40-7 

20 

1-00 

2-54 

170 

17'10 

43-4 

25 

1-30 

3-3 

180 

18-15 

46-1 

30 

1-625 

4-1 

190 

19-20 

48-8 

35 

40 

2-00 
2-45 

5-1 

6-2 

200 
210 

20-25 
21-30 

51-4 
54-1 

45 

2-95 

7-5 

220 

22-35 

56-8 

50 

3-55 

9-0 

230 

23-40 

59-4 

60 

4-65 

11-8 

240 

24-45 

62-1 

70 

5-85 

14-9 

250 

25-50 

64-7 

80 

7-10 

18-0 

260 

26-50 

67-3 

90 

8-35 

21-2 

270 

27-50 

69-8 

100 

9-60 

24-4 

280 

28-50 

72-4 

110 

10-75 

27-3 

290 

29-50 

74-9 

120 

11-85 

30-1 

300 

30-50 

77-4 

130 

12-90 

32-8 

Failing 
cases  in 
practice 


When  spark-gaps  between  needle  points  are 
used  to  measure  very  low  voltages  very  un- 
satisfactory results  are  obtained.  Even  when 
the  electrodes  are  spherical  it  is  very  difficult  to  obtain 
consistent  results  when  the  distance  between  them  is  less 
than  a  millimetre.  When  the  electrodes  are  at  microscopic 
distances  apart  the  above  formulae  cannot  be  applied  in 
practice.  G.  M.  Hobbs  (Phil.  Mag.  [6],  10,  p.  617)  has 
shown  that  when  the  minimum  distance  x  between  the 


180        ELECTRIC  CABLES   AND   NETWORKS 


spheres  is  less  than  3^,  where  ^  =  10~6  metre,  the  spark- 
ing potentials  are  practically  independent  of  the  nature  of 
the  gas  between  the  electrodes.  They  depend,  however, 
on  the  metal  of  which  the  electrodes  are  made.  When 
the  electrodes  are  very  close  together,  it  has  to  be  remem- 
bered that  our  assumption  of  an  isotropic  medium  bounded 
by  smooth  rigid  equipotential  surfaces  is  no  longer  per- 
missible. If  the  surfaces  were  magnified  sufficiently  they 
would  be  seen  to  be  rough,  and  the  dielectric  surrounding 
the  microscopic  projections  would  probably  be  ionized. 
In  these  circumstances,  therefore,  accurate  calculations 
would  be  difficult. 

Hence,  in  determining  dielectric  strengths,  it  is  necessary 
to  have  the  electrodes  at  appreciable  distances  apart,  and 
therefore  high  voltages,  must  be  used.  It  is  not  safe  to 
calculate  dielectric  strengths  from  the  observed  disruptive 
voltages  when  the  electrodes  are  less  than  a  millimetre 
apart.  When  a  maximum  inaccuracy  of  more  than  1  per 
cent,  is  not  permissible,  they  should  be  at  least  half  a 
centimetre  apart. 

It  has  also  to  be  remembered  that  the  formulae  for  the 
maximum  value  of  the  electric  stress  on  the  medium  be- 
tween spherical  electrodes  have  been  obtained  on  the 
assumption  that  the  Faraday  tubes  are  in  statical  equili- 
brium. In  the  case  of  impulsive  rushes  of  electricity  (see 
Chapter  XII),  or  with  alternating  pressures  at  exceedingly 
high  frequencies,  the  disruptive  voltages  seem  to  be  inde- 
pendent of  the  shape  of  the  electrodes. 

The    dielectric    strength    of    a    gas    may    be 

the          deduced    from    experiments    on    the    sparking 

strength  of     voltages    between    spherical    electrodes.      The 

containing  vessel  for  the  gas  should  be  large 

with  the  spherical  electrodes  near  the  centre.     The  dia- 


DIELECTRIC   STRENGTH  181 

meter  of  the  supporting  rods  should  be  small  compared 
with  the  diameter  of  the  electrodes,  and  care  should  be 
taken  that  no  conducting  materials  or  insulating  materials 
having  dielectric  coefficients  different  from  the  gas  are  in 
the  immediate  vicinity  of  the  electrodes,  otherwise  the 
distribution  of  the  Faraday  tubes  between  the  electrodes 
will  be  altered  and  our  formulae  will  not  apply.  It  is 
usually  best  to  earth  the  middle  point  of  the  secondary 
coil  of  the  transformer,  or  the  middle  point  of  the  batteries 
used,  so  as  to  make  the  potentials  of  the  electrodes  equal 
and  opposite  at  the  instant  of  discharge. 

If  E/2  and  —  E/2  be  the  potentials  of  the  electrodes, 
at  the  instant  of  discharge,  when  direct  voltages  are  used, 
we  have 


where  Rmax.  is  the  dielectric  strength  of  the  gas,  x  the  mini- 
mum distance  between  the  electrodes,  and  /  a  number 
which  can  be  obtained  from  Table  I.  The  nearest  points 
on  the  electrodes  should  not  be  closer  than  about  half  a 
centimetre,  and  their  diameter  should  be  about  5  cm. 
With  air  at  atmospheric  pressure  a  voltage  slightly  less 
than  20  kilovolts  would  be  required  when  x  was  0*5  cm. 

When  alternating  pressures  are  used  it  is  absolutely 
necessary  to  know  the  ratio  of  the  maximum  voltage  E 
to  the  effective  voltage  V.  Let  this  ratio,  which  is  some- 
times called  the  amplitude  factor,  be  denoted  by  k,  then 
our  formula  is 


Steinmetz's  method  of  putting  the  electrodes  into 
nitrate  of  mercury,  and  rubbing  them  with  a  clean  cloth, 
before  and  during  the  experiments  is  to  be  commended. 
This  is  especially  necessary  when  the  electrodes  are  only 
a  small  distance  apart. 


182        ELECTRIC  CABLES  AND  NETWORKS 

The  pressure,  temperature,  and  humidity  of  the  gas 
must  be  given. 

J.  N.  Collie  and  W.  Ramsay  (Proc.  Roy.  Soc.,  vol.  lix., 
p.  257,  1896)  give  interesting  comparative  values  of  the 
sparking  potentials  for  various  gases  contained  in  glass 
tubes.  The  electrodes  were  of  platinum  with  slightly 
rounded  ends.  Owing  to  the  dielectric  coefficient  of  the 
glass  tube  being  different  from  that  of  the  gas,  and  owing 
to  the  great  electric  stress  at  the  electrodes  causing  ex- 
cessive ionization,  absolute  values  cannot  be  found  from 
their  results,  but  the  following  table  shows  that  the  di- 
electric strengths  of  the  gases  differ  considerably  :— 


Sparking 
Gas.  Distances  in  mms. 

Oxygen 23 

Air 33 

Hydrogen 39 

Argon       ......         45'5 

Helium     ....    greater  than  250. 


The  dielectric  strength  of  helium,  therefore,  is  extra- 
ordinarily low  compared  with  that  of  other  gases. 

The  liquid  to  be  tested  is  generally  placed 
Dielectric 
strength  of     in  a  vertical  glass  cylinder  about  2  in.  in  dia- 

liquids 

meter.  Spherical  electrodes  about  half  an  inch 
in  diameter  are  immersed  in  the  liquid,  and  the  distance 
between  them  is  varied  by  means  of  a  micrometer  screw. 
The  formulae  for  deducing  the  dielectric  strength  from  the 
disruptive  voltage  are  the  same  as  for  a  gas. 

The  electrodes  should  not  be  less  than  0-3  of  a  centi- 
metre apart,  and  at  this  distance  40  or  50  kilovolts  will  be 
required  to  break  down  good  insulating  oils.  In  some 
cases  when  water  is  present  much  smaller  voltages  suffice. 

In  order  to  find  the  true  dielectric  strength  of  an  oil,  it 


DIELECTRIC   STRENGTH  183 

is  necessary  to  thoroughly  dry  it  before  the  test.  This 
can  be  done  by  letting  hot  air  bubble  up  through  it.  It  is 
inadvisable,  however,  to  heat  the  oil  above  100°  C.  as  con- 
siderable discolouration  often  results  and  its  physical  state 
alters.  When  oils  are  dried  in  this  way  perfectly  con- 
sistent results  can  be  obtained. 

As  a  numerical  example,  let  us  suppose  that  the  dis- 
ruptive voltage  for  an  oil  between  1  cm.  spherical  electrodes, 
0*3  of  a  centimetre  apart,  is  28  kilovolts,  V\  being  equal 
to  — F2,  and  the  amplitude  factor  being  1-5.  By  Table 
I,  we  get 

Bwafl.=(l-5x28/0-3)xl-21 

=  168  kilovolts  per  centimetre. 
If   the   spherical   electrodes   can   be   entirely 
Btrength^of    em^edded  in  the  insulating  material  then  we 

iSg*JJJj°      can  proceed  as  for  liquids  and  gases,  the  same 
formulae  being  employed. 

The  method  frequently  adopted  of  putting  thin  sheets 
of  the  insulating  material  between  metal  electrodes  in  air 
is  of  doubtful  value.  As  the  voltage  is  increased  the  air 
surrounding  the  electrodes  is  broken  down  long  before 
the  disruptive  voltage  is  reached.  The  insulating  material 
heats  excessively,  and  the  maximum  electric  stress  to  which 
it  is  subjected  cannot  be  calculated  as  the  temperature  is 
rarely  uniform  throughout,  and  the  insulativity  of  the 
medium  and  the  dielectric  coefficient  vary  with  the  tem- 
perature. Results  obtained  by  neglecting  the  variations 
of  the  electric  stress  due  to  temperature  are  useful  only 
when  all  the  conditions  of  the  experiment  are  mentioned. 

Dielectric         When   the   insulating   material   is   eolotropic 

Seoeio?ropicf    ^e  GS^CU^^OU  °f  tne  electric  stresses  is  very 
solids        difficult.     They    vary    with    the    dielectric    co- 
efficients and  the  insulativities  of  the  various  constituents, 


184       ELECTRIC  CABLES  AND  NETWORKS 

and,  as  we  have  just  mentioned,  these  quantities  vary 
rapidly  with  the  temperature.  Accurate  measurements  of 
the  mean  dielectric  strength  are  therefore  in  many  cases 
almost  impossible. 

REFERENCES, 

H.  W.  Turner  and  H.  M.  Hobart,  The  Insulation  of   Electric  Ma- 

chines. 
A.   Schuster,  "  The   Disruptive   Discharge    of    Electricity  through 

Gases."     Phil  Mag,  vol.  xxix,  p.  192,  1890. 
C.  P.  Steinmetz,  "  Note  on  the  Disruptive    Strength  of  Insulating 

Materials."     Trans.  Amer.  Inst.  Elect.  Eng.,  vol.  x,  p.  85,  1893. 
T.  Gray,  "  The  Dielectric  Strength  of  Insulating  Materals."     Phys. 

Rev.,  vol.  vii,  p.   199,  1898. 
C.  P.  Steinmetz,     "Dielectric  Strength  of  Air."     Trans.  Amer.  Inst. 

Elect.  Eng.,  vol.  xv,  p.  281,  1898. 
C.  E.  Skinner,     "  Energy  Loss  in  Commercial  Insulating  Materials 

when  Subjected  to  High  Potential  Stress."     Trans.  Amer.  Inst. 

Elect.  Eng.,  vol.  xx,  p.  1047,  1902. 
E.  Jona,  "  Dielectric  Strength  of  Air,  Oil  and  Various  Liquids."    Atti 

deW  Associazione  Elet.  Italiana,  vol.  vi,  p.  3. 
H.  J.  Ryan,  "  The  Conductivity  of  the  Atmosphere  at  High  Voltages." 

Trans.  Amer.  Inst.  Elect.  Eng.,  vol.  xxiii,  p.  101,  1903. 
E.  Jona,  "  Limits  of  High  Tension  Tests  on  Polyphase  Apparatus." 

Elettricista,  Rome,  vol.  13,  p.  113,  1904. 
A.  Russell,  "  The  Dielectric  Strength  of  Air."     Phil.  Mag.  [6],  vol.  ii, 

p.  258,  1906. 
A.  Russell,   "  The    Dielectric    Strength    of    Insulating    Materials." 

Journ.  of  the  Inst.  of  Elect.  Eng.,  vol.  40,  p.  6,  1907. 
R.    P.    Jackson,  "  Potential     Stresses    as    affected    by    Overhead 

Grounded  Conductors."     Trans.  Amer.  Inst.  EL  Eng.,  vol.  26, 

p.  435,  1907. 

H.  W.  Fisher,  "  Spark  Distances."     High  Tension  Power  Trans- 
mission.    St.  Louis  El.  Cong.,  vol.  2,  p.  91,  1904. 
H.  J.  Ryan,  "  Some  Elements  in  the  Design  of  High  Pressure  Insu- 
lation."    High   Tension    Power    Transmission.      St.  Louis   EL 

Cong.,  vol.  2,  p.  278,  1904. 


THE  GRADING  OF  CABLES 


CHAPTER  IX 

The   Grading    of  Cables 

The  grading  of  cables— Concentric  main— Suitable  dimensions  for 
a  concentric  main — Grading  single  core  cables  for  alternating 
pressures— Grading  single  core  cables  for  direct  pressures— 
Jona's  graded  cables — The  effects  of  leakage  currents  on  the 
grading  of  concentric  mains — Numerical  example — The  British 
standard  radial  thicknesses  for  jute  and  paper  dielectrics — 
The  effects  of  stranding  on  the  electric  stresses — Conclusions — 
References. 

The  grading  ^Y  ^e  grading  of  cables  is  meant  the  arranging 
of  cables  of  ^he  or(jer  anc[  thickness  of  properly  chosen 
insulating  wrappings  so  that  each  bears  its  due  share  of  the 
total  electric  stress  to  which  the  insulating  material  is  sub- 
jected. In  addition,  the  electric  stress  on  every  wrapping 
must  be  as  uniform  as  possible  throughout  its  substance. 
We  shall  see  that  it  is  only  possible  to  secure  absolute 
uniformity  of  stress  in  a  wrapping  by  making  the  dielectric 
coefficient  of  the  insulating  material  diminish  in  a  regular 
manner  as  we  proceed  outwards  from  the  axis  of  the  cable. 
In  the  special  case  of  a  single  core  main  with  a  homo- 
geneous dielectric,  the  maximum  electric  stress  -fi^.  occurs 
next  the  core,  and  the  minimum  Rmin,  next  the  lead  sheath. 
If  a  be  the  radius  of  the  core,  the  surface  of  which  we  sup- 
pose to  be  smooth,  and  6  the  inner  radius  of  the  lead  sheath, 
we  have  flM.  =  F/aloge(6/a),  and  jRmin.  =  F/6  loge(6/a), 
where  F  is  the  potential  difference  between  the  core  and 
the  lead  sheath.  We  see  that  in  this  case  Emax>/Rmin=b/a, 


"mm. 
187 


188        ELECTRIC  CABLES  AND  NETWORKS 

and  if  b/a  be  large,  the  material  next  the  core  will  have  to 
withstand  a  stress  much  greater  than  the  average  stress, 
which  equals  V/(b  —  a).  The  layer  of  insulating  material, 
therefore,  has  to  be  made  very  much  thicker  than  if  it  had 
merely  to  insulate  from  one  another  two  infinite  plane 
surfaces  at  the  given  potentials.  If  it  were  possible  to 
make  the  electric  stress  on  the  dielectric  of  a  single  core 
main  uniform  throughout  so  that  its  value  was  V/(b  —  a), 
the  thickness  of  the  layer  would  be  considerably  reduced, 
and  a  considerable  economy  could  be  effected  by  the  smaller 
amount  of  sheathing  and  armouring  required. 

We  shall  show  how  this  can  be  done  by  using  insulating 
materials  having  different  dielectric  coefficients  and  arranged 
in  concentric  wrappings  of  suitable  thicknesses,  and  in  a 
given  order.  Before  doing  this,  however,  we  shall  discuss 
the  formula  R  =  V  /  {  xloge(b/a)  }  for  a  concentric  main,  where 
R  is  the  electric  stress  at  points  the  distance  of  which  from 
the  axis  of  the  cable  is  x.  A  proof  of  this  formula  is  given 
in  the  author's  treatise  on  Alternating  Currents,  vol.  i,  p.  95. 
Concentric  ^he  formula  shows  us  that  the  value  of  R 
at  a  point  in  the  dielectric  is  independent  of 
the  absolute  values  of  the  potentials  of  the  mains,  and 
depends  merely  on  the  difference  of  the  potentials  and  the 
distance  of  the  point  from  the  axis.  We  obviously  have 


If  V  and  b  remain  constant  — 

V 


da  -{«log,(&/«)}*{1-10g'(6/0)K 
Hence,  if  a  be  less  than  b/e  where  e  is  the  base  of  Neperian 
logarithms,  jR^  will  diminish  as  a  increases.  In  this  case, 
we  see  that  the  breaking  down  of  the  dielectric  round  the 
inner  core  actually  diminishes  the  maximum  stress  to  which 
the  dielectric  is  subjected.  It  is  only  when  the  radius  of 


•        THE   GKADING   OF  CABLES  189 

the  charred  dielectric  gets  greater  than  b/e  that  a  disruptive 
discharge  ensues. 

Jona  (Trans.  Int.  Cong.,  St.  Louis,  vol.  ii,  p.  550)  describes 
an  experiment  on  the  disruptive  voltages  of  two  single-core 
cables  of  very  different  diameters,  but  each  wound  with  the 
same  thickness  (1-4  cm.)  of  paper  insulation.  The  core  of 
one  consisted  of  a  thin  wire  O'l  cm.  in  diameter,  while  the 
other  was  a  copper  cylinder  2-9  cm.  in  diameter.  The 
former  broke  down  at  40  kilo  volts,  and  the  latter  at  from 
75-80  kilo  volts.  The  former  also  got  exceedingly  hot 
after  being  subjected  to  30  kilo  volts  for  an  hour,  whilst  the 
latter  was  still  cold  after  50  kilovolts  had  been  applied 
for  the  same  time.  If  we  calculate  the  maximum  electric 
stress  on  the  dielectric  surrounding  the  thin  wire,  on  the 
assumption  that  no  part  of  it  is  broken  down  before  the 
disruptive  discharge  ensues,  we  get  — 


""**•"  0'05loge(145/0-05) 

=  238  kilovolts  per  centimetre. 

Similarly,  the  experimental  results  with  the  thick  cable 
make  -R^oa..  lie  between  76-5  and  81*6  kilovolts  per  centi- 
metre. This  experiment  is  quoted  by  Jona  to  show  that 
the  ordinary  formula  cannot  be  applied  when  b/a  is  large. 
If,  however,  we  assume  that  the  disruptive  discharge 
does  not  occur  until  the  outer  radius  of  the  charred  dielectric 
becomes  equal  to  b/e,  the  experiment  on  the  thin  wire 
gives  us  — 

_40  x  2-718 
**».-        145 

=75  kilovolts  per  centimetre,  nearly, 

which,  being  in  substantial  agreement  with  the  results 
given  by  the  test  on  the  thick  cable,  is  a  striking  confirma- 
tion of  the  theory  outlined  above. 


190        ELECTRIC  CABLES  AND  NETWORKS 

Let  us  suppose  that  the  maximum  working 

dimensions     voltage    F,   the   density  of  the  current  in  the 

concentric     inner  conductor,  and  the  maximum  permissible 

stress  to  which  the  dielectric  may  be  subjected, 

are  fixed.     Let  us  first  suppose  that  the  inner  cylindrical 

conductor  is  solid  and  that  its  radius  is  a.     If,  then,  V/l  be 

the  maximum  permissible  stress,  we  have 

F  F 

aloge(b/a)        I  ' 
and  thus, 

b  =  ael/a. 
Hence  also, 


da          \       a 

If,  therefore,  a  be  greater  than  I,  d  b/d  a  is  positive,  and 
therefore  b  increases  as  a  increases,  but  if  a  be  less  than  Z, 
b  diminishes  as  a  increases.  In  the  latter  case  it  Would 
obviously  be  advantageous  to  make  the  inner  conductor 
hollow,  its  section  remaining  constant,  so  as  to  increase 
the  value  of  a  and  diminish  the  value  of  b.  The  quantity 
of  armouring  and  insulating  material  used  would  be 
diminished  by  this  procedure.  We  conclude,  therefore, 
that  if  a  solid  inner  conductor  of  the  required  cross-section 
would  have  a  radius  less  than  I,  the  inner  conductor  should 
be  made  hollow  and  its  outer  radius  should  not  be  less  than 
I.  In  some  cases  it  would  be  advantageous  to  make  the 
inner  conductor  of  aluminium. 

Although  the  inner  radius  of  the  outer  conductor  begins 
to  increase  when  a  gets  greater  than  I,  the  following  reason- 
ing shows  that  the  quantity  of  the  dielectric  required 
diminishes  until  a  gets  greater  than  1-25  L 

Using  the  same  notation,  the  area  of  the  cross-section 
of  the  dielectric  of  the  cable  is  TT  (b2  —  a2),  and  we  have  to 


THE   GKADING  OF  CABLES  191 

find  the  value  of  a  that  makes  a2  (e2l/a — 1)  a  minimum. 
Differentiating  with  respect  to  a  and  equating  to  zero,  we  get 

ezlfa=a/(a—l). 
Let  a  =  nl,  then 

2/n  =  logen—loge(n—l). 

By  trial,  we  find  that  ft,  =  1*2550  .  .  .  satisfies  this  equation, 
and  hence,  when  n  has  this  value  the  quantity  of  insulating 
material  required  is  a  minimum.  In  this  case  a  =  1-2551, 
6=2'784Z,  and  b=2'2lSa.  As  the  saving  of  insulating 
material  effected  by  increasing  a  from  Z  to  1*25  I  is  only 
about  3  per  cent,  it  is  of  little  importance  compared  with 
the  increased  cost  of  the  armouring. 

We  conclude,  therefore,  that  high-pressure  concentric 
cables,  having  isotropic  dielectrics,  for  use  at  a  maximum 
voltage  V,  should  be  constructed  so  that  b  =ael/a,  where 
V/l  is  the  maximum  permissible  working  stress  to  which 
the  dielectric  may  be  subjected,  and  a  should  never  be 
made  less  than  I. 

We  shall  first  make  the  supposition  that  all 
single  core     the  insulating  wrappings  used  have   the  same 
alternating     dielectric  strength,  and  that  the  maximum  and 
minimum  stresses  to  which  they  are  subjected, 
when  working,  are  to  be  the  same  for  them  all.     We  shall 
also  suppose  that  the  leakage  current  across  the  dielectric 
can  be  neglected  in  comparison  with  the  capacity  current. 
Let  us  suppose  that  there  are  n  insulating  wrappings  the 
inner  radii  of  which  are  a,  r2,  r3,  ...  rn,  respectively,  where 
a  is  the  outer  radius  of  the  lead  tube  encasing  the  inner  core, 
and  let  b  equal  the  inner  radius  of  the  lead  sheath.     Since 
the  ratio  of  the  maximum  to  the  minimum  electric  stress 
is  to  be  the  same  in  all  the  wrappings,  we  must  have 
r2       r3  b 


192       ELECTRIC   CABLES  AND  NETWORKS 

We  see,  therefore,  that  the  radii  should  be  in  geometrical 
progression,  the  common  ratio  being  (b/a)l/n.  The  thick- 
nesses of  the  layers  also  form  a  geometrical  progression 
having  the  same  ratio  (b/a)yn. 

Let  Fi,  F2,  .  .  .  Fw+1,  be  the  potentials  of  points  at  dis- 
tances a,  r2,  .  .  .  b  from  the  axis  of  the  cable.  Then,  since 
the  layers  form  n  condensers  in  series,  the  potential  difference 
across  a  layer  will  be  inversely  proportional  to  the  capacity 
of  the  layer,  and  thus  we  have 

Fi—  F,  F2-F3  Fn-Fn+1 

(lAi)log.(ra/a)      (l/Xa)loge(r3/ra)  UAn)loge(&/rj' 

Hence,  since  the  P.D.s  are  all  in  phase,  each  of  these  ratios 
equals  V{S(l/*>J  loge  (rn+1/rj}9  where  F  is  the  voltage 
applied  between  the  core  and  the  sheath. 

If  Rm  denote  the  maximum  electric  stress  on  the  rath 
layer,  we  have 

-p     _        ' 


F      -  F 

'  m  v  m  +  1 


=(  V/\mrm)/S(l/\m)loge(rm+l/rn). 

Now,  since  the  maximum  stress  on  every  layer  is  to  be 
the  same,  we  must  arrange  so  that 

\1a 
Therefore 


a 

Hence  \i,  X2,  ...  \,  are  the  terms  of  a  geometrical  progression 
whose  common  ratio  is  (b/a)1/n-. 

If  Rmax,  denotes  the  maximum  electric  stress  in  the  graded 
cable,  we  have 


-^1- 

~  a  li 


b/ct — 1       T      /7  /  \ 
1/n_1}log  (6/ft) 


=  Rfmax 

b/a—l 


THE   GKADING   OF  CABLES 


193 


where  R'max.  stands  for  V/a  log  (b  /a)  the  maximum  stress  in 
a  cable  of  the  given  dimensions  with  an  isotropic  dielectric. 
If  Rmin,  denote  the  minimum  electric  stress  in  the  dielectric 
of  the  graded  cable,  We  have 


In  the  ideal  cable  n  would  be  infinite,  and  thus  the 
stress  would  be  the  same  at  all  points,  and  would  equal 
V/(b-a). 

The  capacity  per  unit  length  of  a  single-core  cable,  with 
isotropic  dielectric  equals  X/  {  2loge(b/a)  }  .  The  capacity  of  the 
graded  cable  equals  \1n{(b/a)ln—  1  }/{(b/a—  l)21oge(6/a)}, 
When  n  is  infinite  this  equals  X1a/{2(6  —  a)}.  If  X  be 
the  dielectric  coefficient  of  the  cable  with  the  isotropic 
dielectric,  and  Xmax  be  the  dielectric  coefficient  of  the  inner 
coating  of  a  graded  cable  having  n  layers,  the  capacities  of 
the  cables  will  be  equal  if 

>W  =X(6/*-l)/»  {  (&/«)'  "-1  }  . 

If  the  value  of  X,^  be  less  than  this,  the  capacity  of  the 
graded  cable  will  be  the  smaller.  For  example,  if  there  are 
4  layers  and  b/a  equals  3,  we  find  that  the  capacities  are 
equal  when  Xmaa.  —  1-58  X.  In  this  case  the  minimum  value 
of  X  in  the  graded  cable  is  0-69  X. 

To  illustrate  how  the  value  of  the  maximum  electric 
stress  diminishes  as  the  number  of  wrappings  is  increased 
we  shall  work  out  a  few  numerical  examples. 

(i)  Two  wrappings  (n=2)  — 


b/a 



2 

3 

4 

5 

Rmas 

/Rmin  graded  dielectric     . 

1-414 

1-732   ;       2 

2-236 

X\i   m« 

x./R/min.  isotropic  dielectric    . 

2 

3 

4. 

5 

J^max 

yil'max    

0-828 

0-732   !   0-667 

0-618 

Per 

cent,    increase    of    the    per- 

missible  voltage  due  to  grading 

21 

37 

50 

62 

194        ELECTRIC  CABLES   AND  NETWORKS 

(ii)  Three  wrappings  (n=3)— 


b/a   

2 

3 

4 

5 

Rmax./Rmh,.  graded  dielectric    . 

T260 

1-442 

1-587 

1-710 

R'max  /R'min  isotropic  dielectric     . 

2 

3 

4 

5 

R       /R' 

0-780 

0-663 

0-587 

0-532 

Per    cent,    increase    of    the    per- 

missible voltage  due  to  grading 

28 

51 

70 

88 

(iii)  Four  wrappings  (n  =  4) — 


b/a   

2 

3 

4 

5 

Rmax./Rmin.  graded  dielectric    . 

1-189 

1-316 

1-414 

1-495 

R'max./R'min.  isotropic  dielectric    . 

2 

3 

4 

5 

Rmax  /R'jnk  

0-756 

0-632 

0-552 

0-495 

Per    cent,    increase    of    the  per- 

missible voltage  due  to  grading 

32 

58 

81 

102 

(iv)  Ideal  uniformly  graded  cable  (r&=infinity) — 


b/a   

2 

3 

4 

5 

Rmax./R'max.            

0-693 

0-549 

0-462 

0-402 

Per    cent,    increase    of    the    per- 

missible voltage  due  to  grading 

44 

82 

116 

149 

We  have  assumed  above  that  the  dielectric  strengths  of 
all  the  insulating  wrappings  are  the  same.  If,  however, 
the  dielectric  strengths  are  known  accurately  and  are  not 
all  the  same,  another  solution  may  be  preferable.  If  Rm 
be  the  safe  working  stress  for  the  mth  layer,  we  have 


Since  it  is  advisable  to  make  the  ratio  Rm(Kc./Rmin.  the  same 
for  all  layers,  the  ratio  rm+l/rm  will  be  constant,  and  as 
before  r2,  r3,  ...  rn  will  be  the  n  —  1  geometrical  means 
between  a  and  b. 

The  above  equation  shows  that  we  must  have 


THE   GKADING   OF   CABLES  195 

Since  a,  r2t  ...  rn,  are  in  an  ascending  order  of  magnitude, 
Xt  EI,  A  2  E2,  ...  \En,  must  be  in  a  descending  order.  We 
see,  therefore,  that  it  is  necessary  to  put  the  wrappings 
whose  constants  are  (X,  E)  over  the  wrapping  whose  con- 
stants are  (X',  R'),  if  X  E  be  greater  than  X'  R',  even  although 
X'  may  be  less  than  X. 

When,  however,  the  main  object  we  have  in  view  is  to 
make,  at  all  costs,  the  factor  of  safety  of  the  cable  as  high 
as  possible,  it  is,  in  general,  advisable  to  put  the  insulating 
material  having  the  greatest  dielectric  strength  in  contact 
with  the  core,  and,  if  possible,  grade  the  dielectric  by  using 
outer  layers  having  smaller  dielectric  coefficients. 

At  the  moment  of  switching  on   the   direct 

Grading 

single  core     pressure,  the  distribution  of  the  electric  stresses 

cables  for 

direct          depends    on    the    dielectric    coefficients    of    the 
pressures 

wrappings,  but  after  a  few  seconds,  when  the 

leakage  current  attains  its  steady  value,  the  values  of  the 
stresses  depend  on  the  insulativities  of  the  various  wrappings. 
If  the  pressure  be  always  applied  gradually  at  the  start  we 
may  neglect  the  dielectric  coefficients  and  grade  the  cable 
for  the  steady  pressure  taking  the  insulativities  only  into 
account. 

Let  o-j,  o-2,  ...  be  the  insulativities  of  the  various  wrappings 
and  C  the  leakage  current,  then  (p.  51)  the  resistance  of 
the  mth  cylinder  to  the  flow  of  the  current  C  across  it  is 


Hence  we  have, 


c=,     F-F°      = 


(o-1/2?r)log€(r2/a)     (o-2/27r)loge(r3/r2) 
and  therefore, 


y 

n        Y n+l 


196        ELECTRIC   CABLES   AND   NETWORKS 

Also  for  the  mth  layer 

d  v 


For  reasons  stated  above,  we  choose  the  radii  of  the  bound- 
aries between  the  wrappings  so  that  they  are  the  n — 1 
geometric  means  between  a  and  b.  Hence,  if  the  factor 
of  safety  is  to  be  the  same  for  all  the  layers,  we  must  have 

rmRm/<rm=  constant. 
If  the  dielectric  strengths  are  all  equal,  this  simplifies  to 

rm/o-m—  constant. 

Hence,  proceeding  as  in  the  corresponding  problem  for 
alternating  pressures,  we  see  that  the  economies  that  can 
be  effected  by  suitably  grading  the  insulativities  of  the 
various  wrappings  are  the  same  in  the  two  cases.  It  has 
always  to  be  remembered,  however,  that  the  values  of  the 
insulativities  of  insulating  materials  vary  with  temperature 
much  more  rapidly  than  their  dielectric  coefficients.  The 
following  table  given  by  A.  Campbell  (Proc.  Roy.  Soc.  A., 
vol.  Ixxviii,  p.  196)  illustrates  the  effects  of  temperature 
on  the  physical  properties  of  oven-dried  cellulose. 


Temperature 
Centigrade. 

Dielectric 
Coefficient. 

Insulativity 
106  Megohm-cm. 

40 

6-7 

25 

— 

1,600 

30 

6-8 

900 

40 

7-0 

330 

50 

7-2 

125 

60 

7-3 

40 

65 



20 

70 

7-5 

— 

((    UNIVEPxSITY    1 


THE   GRADING  OF  CABLES  197 

The  extremely  rapid  manner  in  which  the  insulativity  of 
cellulose  varies  with  the  temperature  is  typical  of  the 
behaviour  of  the  other  insulating  materials  used  for  cables. 
Hence  in  connexion  with  the  grading  of  cables  the  heating 
of  the  dielectric  has  to  be  considered.  We  shall  consider 
this  point  in  the  next  chapter. 

Messrs.  Pirelli  &  Co..  of   Milan,  made  experi- 
Jona's 

graded        ments  on  the  grading  of  cables  as  early  as  1898. 
cables 

E.  Jona  (Trans.  Int.  Cong.,  St.  Louis,  vol.    ii, 

p.  550)  has  constructed  single  core  cables  the  insulating 
wrappings  of  which  are  arranged  so  that  those  nearer  the 
core  have  greater  dielectric  coefficients  than  those  more 
remote.  The  layers  next  the  core  are  generally  of  rubber, 
and  round  them  are  wound  layers  of  paper  or  jute  having 
smaller  dielectric  coefficients.  The  more  costly  rubber 
insulation  is  thus  concentrated  where  its  high  dielectric 
coefficient  partially  relieves  the  excessive  electric  stress,  and 
its  great  dielectric  strength  enables  it  to  withstand  easily 
this  diminished  stress.  The  value  generally  accepted  for 
the  dielectric  strength  of  pure  vulcanized  para  is  15-20 
effective  kilovolts  per  millimetre,  or  20-30  direct  kilovolts 
per  millimetre. 

According  to  Jona,  the  value  of  the  dielectric  coefficient 
X  of  pure  vulcanized  rubber  is  3.  We  can  increase  the  value 
of  X  without  appreciably  weakening  the  dielectric  strength 
by  "  loading  "  it  with  certain  materials.  The  following  data, 
taken  from  Jona's  paper  (I.e.  ante)  illustrate  that  X  can 
easily  be  varied  throughout  wide  limits. 


58  per  cent,  para,  26  per  cent,  talc,  14  per  cent,  oxide  of 

zinc,  2  per  cent,  sulphur 
64  per  cent,  para,  16   per  cent,  talc,  8  per   cent,  sulphur, 

8  per  cent,  minium,  4  per  cent,  oxide  of  zinc 
55  per  cent,  para,  22-2  per  cent,  talc,  22-2  per  cent,  sulphur 


4-4-2 

5 
6-1 


198        ELECTRIC  CABLES  AND  NETWORKS 

The  following  is  a  description  of  a  Jona  graded  cable 
(Fig.  53)  which  successfully  withstood  a  testing  pressure 
of  150  kilovolts  at  the  Milan  Exhibition  (1906).  The  core 
consists  of  nineteen  strands  of  copper  wire,  the  diameter 


FIG.  53. — Jona's  Graded  Cable.     The  stranded  core  is  surrounded  with  a 
closely   fitting   lead   tube.     There   are   four   insulating  layers. 

of  each  of  which  is  3-3  mm.  The  cross-section  of  the  copper 
is  therefore  162  mm.2.  Round  this,  for  reasons  explained 
later,  is  a  close-fitting  lead  tube,  the  outer  diameter  of 
which  is  18  mm.  The  insulation  is  built  up  as  follows— 


Thickness 

A 

in  mm. 

First  layer.     Rubber 

2-5 

6'1 

Second  layer.     Rubber       

2-3 

4-7 

Third  layer.     Rubber  

4-5 

4-2 

Fourth  layer.     Impreg.  paper       .... 

. 

5-2 

4-0 

The  total  thickness  of  the  insulation  is  therefore  14-5  mm. 
(b/a=2-Ql),  and  the  cable  is  lead-covered. 

If  R,  R',  R",  and  R'",  be  the  maximum  electric   stresses 


THE   GRADING   OF  CABLES 


199 


on  the  four  layers  when  the  applied  pressure  is  1 50  kilovolts, 
we  find  by  the  formula  given  on  page  192,  J?  =  124, 
^  =  132,  R"  =  123,  and^'''  =  97-4kilovolts  per  centimetre.  If 
a  dielectric  of  homogeneous  substance  had  been  used,  the 
maximum  electric  stress  would  have  been  174  kilovolts. 
Hence  the  grading  has  reduced  the  maximum  electric 
stress  by  about  24  per  cent.  If  air  had  been  the  dielectric, 
a  disruptive  discharge  would  have  ensued  at  23  kilovolts. 


FIG.  54. — Single  Core  Cable  with  two  homogeneous  insulating  coverings. 

M.  0' Gorman  has  suggested  that  by  suitably  "  loading  " 
paper  insulation  we  might  make  the  electric  stress  almost 
uniform  throughout  the  dielectric.  He  points  out  that  this 
can  be  done  in  a  single  core  main  by  arranging  that  Xr  is 
approximately  the  same  at  all  points. 

In  the  discussion  given  above  of  the  grading 
The  effects 
of  leakage     or  single  core  mains  for  alternating  pressures, 

on  the        we  made  the  supposition  that  the  leakage  cur- 
grading  of  ,. 
concentric     rents  were  negligibly  small  compared  with  the 

capacity  currents.     We  shall  now  consider  how 
the  presence  of  leakage  currents  modifies  our  results.     To 


200        ELECTRIC  CABLES   AND   NETWORKS 

simplify  the  formulae,  let  us  suppose  that  the  dielectric 
consists  of  two  layers  of  different  isotropic  insulating 
materials  at  the  same  temperature  throughout.  Let  aL, 
Xi  be  the  insulativity  in  ohms  and  the  dielectric  coefficient 
of  the  inner  layer  next  the  inner  conductor,  and  let  cr2,  X2 
be  the  corresponding  quantities  for  the  outer  layer.  Let  r 
be  the  radius  of  the  cylindrical  boundary  between  the  two 
wrappings  (see  Fig.  54).  Then  if  RI,  R2  be  the  resistances 
per  unit  length  to  the  flow  of  leakage  electric  currents  across 
them,  and  KI,  K2  be  the  capacities  in  farads  per  unit  length 
of  the  cylindrical  tubes  formed  by  the  wrappings,  we  have 


21oge(r/a)      9  x  1011  2loge(b/r)      9  x  1011 

Now  the  leakage  current  iR  across  an  isotropic  dielectric 
is  in  phase  with  the  P.D.  applied  at  its  boundaries,  and  the 
capacity  current  iK  is  90°  in  advance  of  this  P.D.  If  v' ', 
v,  and  v"  denote  the  instantaneous  values  of  the  potential 
of  the  inner  conductor,  the  boundary  between  the  two 
media,  and  the  outer  conductor  respectively,  we  have 
-v  v — v" 


R  — 


*"* =K'  ~<n(v'~v} '  *'* =K*T 

where  IR,  i"R  denote  the  leakage  currents,  and  i'K,  i"K  the 
capacity  currents  in  the  inner  and  outer  wrappings  re- 
spectively. We  also  have 

«•'    i  »•'    t"  ji.;" v 

*  JR   I  *  JE  —  *  JB   I  *  K  —  *i 

since  the  sum  of  the  leakage  and  capacity  (displacement) 
currents  in  each  medium  must  equal  the  total  current 
flowing  across  the  dielectric. 

Let   Fi     F2  denote  the  effective  values  of  v' — v  and  of 


THE   GRADING  OF  CABLES  201 

v  —  v",  and  let  (/>i  and  <£2  denote  respectively  the  phase- 
difference  between  Fi  and  F2  and  the  effective  value  of  *. 
Then  if  /  be  the  frequency,  and  a>=2  TT/,  we  have 

tan  <f>i  =  uKiRi  =/\i<n/(18  x  1011),) 
and  tan  <£2  =  a)K2B2=f\2<r2/(l8  xlO11)./  ' 

If,  therefore,  \1a-i=\2  o-2,  then  V\  and  F2  are  in  phase  with 
one  another,  and  thus 


where  F  is  the  effective  value  of  the  applied  P.D.  In  general, 
however,  \icr±  is  not  equal  to  X2<72,  and  therefore  Fi  +  F2 
must  be  greater  than  F. 

Now  by  reciprocating  the  well-known  formula  (see  the 
author's  Alternating  Currents   vol.  i,  p.  166) 


A          {( 
for  the  currents  in  a  divided  circuit,  we  get 

' 


V 

the  formula  for  the  voltages  across  leaky  condensers  in 
series. 

By  differentiating  this  expression  with  respect  to  &>,  it  is 
easy  to  see  that  FI  increases  with  w  when  X2o-2  is  greater 
than  \!o-i.  In  this  case,  the  electric  stresses  in  the  medium 
next  the  inner  conductor  increase  as  the  frequency  increases, 
and  the  stresses  in  the  outer  medium  diminish. 

We  see  from  (B)  that,  when  X2<72  is  greater  than  X^i, 
Fi/F  has  its  minimum  value  when  o>  is  zero,  that  is,  with 
steady  pressures,  and  its  maximum  value  when  a>is  infinite, 
that  is,  with  an  alternating  voltage  of  very  high  frequency. 

Similarly  when  X2o-2  is  less  than  X^i,  Fi/Fhasits  maxi- 
mum value  Ei/(Ei-\-E2)  with  steady  voltages,  and  its  mini- 
mum value  K2/(KL-\-K2)  with  alternating  voltages  of  very 
high  frequency. 


202        ELECTRIC  CABLES  AND  NETWORKS 

Numerical  Let  us  assume  that  the  radius  of  the  inner 
example  conductOr  is  1  cm.  (a  =  l)  the  radius  of  the 
boundary  1-5  cm.  (r  — 1-5),  and  the  inner  radius  of  the  outer 
conductor  2*25  cm.  (6=2-25).  Let  us  also  assume  that  for 
the  outer  jute  wrapping,  cr2  =  1012,  X2=2,  and  that  for  the 
vulcanized  rubber  inner  wrapping,  o-^lO16,  Xi=4.  If  the 
direct  voltage  applied  to  the  conductors  be  30,000,  then, 
putting  a>=0  in  (1),  we  find  that 

F!  =30,000,  and  F2=0,  very  approximately. 

Thus  practically  all  the  electric  stress  comes  on  the  rubber. 
Let  us  now  suppose  that  an  alternating  pressure  of  very 
high  frequency  is  applied  between  the  conductors.     In  this 
case,  putting  o>  equal  to  infinity  in  (1),  we  get 

Vi__  X2loge(fr/V)  J_ 

V       Xilogt(r/a)+Xilog.(^A)        3' 

and  thus,  Fi  is  10,000  volts  and  F2  is  20,000.  Hence,  as 
the  frequency  increases  from  0  to  infinity,  FI  diminishes 
from  30,000  to  10,000  volts,  and  F2  increases  from  0  to 
20,000. 

From  (A),  we  see  that 

tan«/>!=2  x!05//9,  and  tan  <£2  =  10//9. 

Hence,  at  ordinary  frequencies,  the  error  made  by  assuming 
that  $j  and  </>2  are  both  90°  is  small.  If  /  is  greater  than  9, 
Fi/F2  is  nearly  equal  to  1/2. 

In  practice,  therefore,  we  see  that  in  the  case  considered 
the  maximum  pressure  across  the  outer  layer  with  alter- 
nating pressures  may  be  very  much  larger  than  when  a 
direct  pressure  is  applied  between  the  conductors,  the  value 
of  which  equals  the  maximum  value  of  the  alternating 
pressure  between  the  conductors.  On  the  other  hand,  the 
electric  stresses  on  the  inner  dielectric  may  be  much  less 
with  the  alternating  pressures. 


THE   GBADING  OF  CABLES 


203 


The  nominal  area  of  the  cross-sections  of  the 
conductors  and  the  radial  thicknesses  (b — a)  of 
the  dielectric  for  concentric  cables  given  in  the 
following  table  are  taken  from  a  report  issued  by 
the  Engineering  Standards  Committee  (E.S.C.) 
in  August,  1904  (p.  8) — 


The  British 

standard 

radial 

thicknesses 

for  jute  and 

paper 
dielectrics 


660  Volts. 

11,  000  Volts. 

8. 

a. 

b  -  a. 

R 

b  -a. 

Rm 

sq.  in. 

0-025 

in. 
0-089 

in. 
0-08 

K.V.  per  mm. 
0-64 

in. 
0-35 

K.V.  per  mm. 
4-3 

0-050 

0-126 

0-08 

0-59 

0-35 

3-7 

0-075 

0-155 

0-08 

0-57 

0-35 

3-3 

o-ioo 

0-178 

0-09 

0-50 

0-36 

3-1 

0-125 

0-199 

0-09 

0-49 

0-36 

3-0 

0-150 

0-219 

0-09 

0-49 

0-36 

2-9 

0-200 

0-253 

0-09 

0-48 

0-36 

2-7 

0-250 

0-282 

o-io 

0-43 

0-37 

2-6 

In  the  above  table,  8  represents  the  cross-sectional  area, 
a  the  radius  of  the  cylindrical  conductor  whose  cross- 
sectional  area  is  S,  b — a  the  thickness  of  the  dielectric  given 
by  the  E.S.C.,  and  Em  the  maximum  working  electric  stress 
when  the  amplitude  factor  of  the  applied  alternating  pres- 
sure is  \/2. 

It  will  be  seen  that  the  electric  stresses  on  the  dielectric 
are  very  different  in  the  high-pressure  cable  from  what  they 
are  in  the  low-pressure  cable,  and  the  dielectrics  in  cables 
of  different  sizes  are  subjected  to  appreciably  different 
stresses. 

In  the  first  five  of  the  high-pressure  cables,  the  dielectric 
surrounding  the  high-pressure  conductor  will  begin  to  be 
broken  down  before  the  disruptive  discharge  takes  place, 
because  in  these  cables  the  ratio  of  b  /a  is  greater  than  e 


204       ELECTRIC  CABLES  AND  NETWORKS 

(2-718).  The  specified  thicknesses,  therefore,  are  not 
economical.  Take,  for  instance,  the  main  in  which  the 
nominal  cross-sectional  area  of  the  conductor  is  0-025  sq.  in. 
With  a  solid  cylindrical  conductor  a  equals  0-089  in.,  and 
b  is,  therefore,  equal  to  0-35+0-089  =0-439  in.  Thus 
&/a=:  4-92.  If  we  make  the  inner  conductor  hollow  and 
a  =0-1 42  in.,  b  =0-3865  in.,  we  get  the  same  maximum  stress 
on  the  dielectric,  but  its  thickness  has  been  reduced  by  33  per 
cent,  and  the  outer  radius  by  12  per  cent.  As  the  armouring, 
etc.,  would  also  be  substantially  reduced,  the  cable  would 
be  less  costly.  If  we  merely  kept  b  =0-439  in.,  but  increased 
a  to  0-1616,  so  that  b/a=e  nearly,  then  the  carrying  capacity 
of  the  cable  would  be  nearly  quadrupled,  the  thickness  of 
the  dielectric  diminished  20  per  cent.,  and  the  maximum 
electric  stress  would  have  been  reduced  to  3-8  kilo  volts  per 
millimetre. 

The  fact  that  the  dielectrics  of  cables  are  not  quite 
isotropic  is  sometimes  advanced  as  a  reason  for  making  the 
radius  of  the  inner  conductor  smaller  than  the  value  indi- 
cated by  theory.  This  practice,  however,  is  founded  on  a 
misapprehension,  as  the  effect  of  diminishing  the  radius  is 
to  increase  the  electric  stress,  and  there  is  no  reason  why 
dielectrics  of  heterogeneous  substance  should  be  subjected 
to  greater  stresses  than  those  of  homogeneous  substance. 
The  want  of  isotropy  may  possibly  be  a  reason  for  increasing 
the  diameter  of  the  inner  conductor,  the  thickness  of  the 
insulating  covering  remaining  the  same. 

In  designing  cables,  it  has  to  be  remembered  that  the 
insulation  resistance,  the  capacity,  the  electric  stresses, 
and,  as  we  shall  see  in  the  next  chapter,  the  thermal  con- 
ductance have  to  be  considered.  For  low-tension  cables, 
the  insulation  resistance  must  be  comparatively  high,  and 
hence,  it  is  not  possible  to  use  too  small  a  value  of  b/a. 


THE   GRADING   OF  CABLES  205 

Similarly,  for  high-tension  cables,  although  a  small  value  of 
b/a  makes  the  electric  stresses  small  and  increases  the 
thermal  conductance  yet  it  makes  the  capacity  large,  and 
the  consequent  large  capacity  current  may  be  a  serious 
drawback  in  practical  work. 
The  effects  In  order  to  simplify  the  formulae  for  the 

of  strand-         ,      ,    .  , 

ing  on  the  electric  stress,  we  have  assumed  that  the  inner 
stresses  conductor  is  a  smooth  cylinder.  In  practice, 
the  inner  conductor  is  nearly  always  stranded,  and  it  is 
necessary  therefore  to  consider  the  effect  of  the  stranding. 
Owing  to  the  greater  curvature  of  the  surface  of  the  strands, 
we  can  see,  from  first  principles,  that  the  effect  will  be  to 
increase  the  maximum  stress.  Jona  found  experimentally 
that  the  brush  discharges  from  solid  wires  and  stranded  or 
braided  wires  having  the  same  external  size  begin  at  prac- 
tically the  same  voltages.  Hence  we  may  infer  that  the 
stranding  of  the  conductor  does  not  much  affect  the 
dielectric  strength  of  the  cable.  It  is  important,  however, 
to  be  able  to  calculate  the  stress  exactly,  and  this  can  be 
done  by  means  of  a  formula  due  to  Professor  Levi-Civita 
(vide  Jona  I.e.  ante).  The  formula  is  given  in  terms  of 
Gauss's  hypergeometric  series,  but  Jona  has  computed  these 
series  for  useful  values  of  the  variables,  so  that  approximate 
solutions  can  be  readily  obtained.  The  results  show  that 
the  effect  of  the  stranding  is  generally  to  increase  the 
maximum  stress  on  the  inner  dielectric  by  about  20  per 
cent.  It  is  worth  while,  therefore,  to  prevent  this  increase 
in  the  stress  on  the  inner  wrapping  by  making  the  sur- 
face of  the  conductor  smooth.  This  can  be  done  by 
covering,  as  Jona  does,  the  inner  conductor  with  a  thin 
lead  tube.  For  extra  high-pressure  cables  the  gain  in  the 
strength  is  well  worth  the  slight  increase  in  the  cost  of 
the  cable. 


206        ELECTRIC   CABLES   AND   NETWORKS 

We  may  sum  up  the  results  arrived  at  in 
Conclusions 

this  and  the  preceding  chapter  as  follows. 

(1)  When  part  of  the  dielectric  under  stress  breaks  down, 
a  disruptive  discharge  ensues  only  when  the  effect  of  this 
partial  breakdown  is  to  increase  the  electric  stress  on  the 
remaining  portion. 

(2)  The  dielectric  strength  of  air  under  given  conditions 
can  be  found  accurately  by  finding  the  disruptive  voltages 
between  spherical  electrodes  at  distances   greater  than  0-5 
of    a    centimetre    apart.     Under   normal    conditions    it    is 
about  3-8  kilovolts  per  millimetre. 

(3)  The  dielectric  strength  of  other  gases  can  be  found 
in  a  similar  way  experimentally  by  the -help  of  the  tables 
given  on  p.   176.       The  dielectric  strengths  of  the  mon- 
atomic  gases  helium  and  neon  are  small. 

(4)  The  dielectric  strength  of  oils  can  be  found  by  noticing 
the  disruptive  voltages   between    spherical  electrodes  im- 
mersed in  them,  provided  that  the  distance  apart  is  greater 
than  0-3  of  a  centimetre.     An  excellent  way  of  drying  oils 
is  by  letting  heated  air  bubble  through  them. 

(5)  In  finding  the  dielectric  strength  of  solids  it  is  ad- 
visable, when  possible,  to  embed  the   spherical   electrodes 
in  the  material  under  test. 

(6)  High-pressure  concentric  cables    having  an  isotropic 
dielectric,  for  a  maximum  working  pressure   F  should  be 
constructed  so  that — 

b=atl/a, 

where  V/l  is  the  maximum  permissible  working  stress  to 
which  the  dielectric  may  be  subjected,  b  is  the  inner  radius 
of  the  outer  conductor,  and  a  is  the  outer  radius  of  the 
inner  conductor.  The  smallest  permissible  value  of  a  is  I. 
When  the  core  is  stranded  it  should  be  encased  in  a  thin 
lead  tube. 


THE   GRADING  OF  CABLES  207 

(7)  With  a  composite  dielectric  subjected  to  alternating 
pressure,  the  P.D.s  across  the  layers  are  usually  out    of 
phase  with  one  another.     It  is  only  in  a  limited  number 
of  cases,  however,  that  the  increase  of  the  stress  due  to  this 
cause  has  to  be  considered,   as  the  leakage  currents  are 
usually  negligibly  small  in  comparison  with  the  capacity 
currents. 

(8)  The   effects   of   alternating   and   direct   pressures   in 
producing   stresses  in  the   dielectric   are   sometimes   quite 
different. 

(9)  High-pressure  cables  for  alternating  or  direct  current 
circuits  should  be  graded  so  as  to  make   the  maximum 
electric  stress  on  the  dielectric  as  small  as  possible,  and 
stranded  conductors  should  be  encased  in  thin  lead  tubes. 

REFERENCES. 

Mervyn  O'Gorman,  "Insulation  on  Cables."     Journ.  Inst.  EL  Eng., 

vol.  xxx,  p.  608,  1901. 
E.  Jona,  "  Insulating  Materials  in  High  Tension  Cables."     Trans. 

Int.  Congress.  St.  Louis,  vol.  ii,  p.  550,  1904. 
A.  Russell,  "  The  Dielectric  Strength  of  Insulating  Materials  and  the 

Grading  of  Cables."     Journ.  Inst.  El.  Engin.,  vol.  40,  p.  6,  1907. 


THE    HEATING  OF    CABLES 


CHAPTER   X 

The   Heating  of  Cables 

The  heating  of  cables — Temperature  gradient  in  a  concentric  main 
—Numerical  example— Effects  of  heat  on  the  dielectric— Effect 
of  the  temperature  gradient  on  the  electric  stress — The  thermal 
conductance  of  a  single  core  main — Numerical  example — The 
temperature  in  the  substance  of  conductors  carrying  uniform 
currents— The  thermal  conductance  of  a  concentric  main— 
The  thermal  conductance  of  polycore  cables — The  heating  of 
bare  conductors — References. 

The  heating  ALTHOUGH  the  rise  of  temperature  in  under- 
of  cables  ground  cables  is  a  problem  of  considerable 
practical  importance,  yet  very  little  information  on  this 
subject  is  available.  It  is  usual  to  consider  that  a  cable  of 
given  dimensions  is  carrying  its  full  load  when  the  current 
density  in  the  core  has  a  given  value.  As  a  rule  the  effect 
of  the  thermal  conductivity  of  the  insulating  wrappings  is 
not  taken  into  account.  The  value  of  this  physical  constant, 
however,  determines  the  difference  of  temperature  between 
the  core  and  the  sheath,  and  hence  the  temperature,  and 
consequently  the  electrical  conductivity  of  the  copper  must 
be  considerably  affected  by  the  value  of  the  thermal  con- 
ductivity of  the  dielectric.  For  high-pressure  cables  also 
the  thermal  gradient  in  the  dielectric  affects,  in  some  cases 
very  seriously,  the  dielectric  strength.  We  shall  therefore 
briefly  consider  the  laws  governing  the  flow  of  heat  across 
the  dielectric  of  cables. 

211 


212        ELECTRIC  CABLES   AND  NETWORKS 

Temperature        When  a  concentric  main  is  carrying  a  current, 
*f*^fji!  In     the  temperature  of  the  dielectric  is  not  uniform 

«i  concen- 

trie  main  owing  to  the  heat  generated  in  the  inner  con- 
ductor. If  the  dielectric  is  isotropic,  the  temperature  at 
any  point  after  the  flow  of  heat  has  become  steady  can  be 
readily  written  down,  if  we  assume  that  the  thermal  con- 
ductivity k  of  the  dielectric  remains  approximately  constant 
over  the  range  of  working  temperatures. 
i  If  6  be  the  temperature  at  all  points  at  a  distance  r  from 
the  axis  of  the  main,  we  have,  since  the  heat  entering  per 
second  an  elementary  cylinder  of  the  dielectric,  coaxial  with 
the  main,  must  equal  the  heat  leaving  it, 


, 
d  r  \          dr  J 

neglecting  the  flow  of  heat  near  the  ends  parallel  to    the 
length. 

Hence  d  0__  __  A 

~dV~     ~Ty 
where  A  is  a  constant.     We  have,  therefore, 


where  02  is  the  temperature  of  the  outer  conductor,  the  inner 
radius  of  which  is  b. 

Let  us  suppose  that  the  inner  conductor,  supposed  of 
copper,  is  solid  and  of  radius  a,  and  that  i  is  the  current 
density  in  it.  Then,  if  p  be  the  volume  resistivity  of  the 
copper  in  ohms,  we  have 

—4-2  x  ZTra 


j 
dr 

and  thus,  A  =  a2i2p/(8-±k). 

Hence  6  =  02+(a2i2p/S-4:  k)  log€(6/r), 

and  61—  0.2  =  (a*i  V/8-4  k)  loge(6/a), 

where  0±  is  the  temperature  of  the  surface  of  the  inner 

conductor. 


THE  HEATING  OF  CABLES  213 

We  have  assumed  above  that  the  thermal  conductivity  k 
of  the  dielectric  does  not  vary  appreciably  with  the  tem- 
peratures likely  to  occur  in  practice.  C.  H.  Lees  (Trans. 
Roy.  Soc.,  p.  433,  vol.  204A,  1905)  has  proved  that  this 
assumption  is  permissible  for  paraffin  wax,  glycerine,  and 
various  other  insulating  materials.  There  appears  to  be 
a  slight  tendency,  however,  towards  lower  conductivity  as 
the  temperature  increases.  G.  F.  C.  Searle  (Proc.  Camb. 
Phil.  Soc.,  xiv,  2,  p.  189,  1907)  has  devised  an  exceedingly 
simple  method  of  determining  the  thermal  conductivity 
of  rubber,  the  value  of  which  he  finds  to  equal  0*0004 
nearly. 

Numerical  To  illustrate  the  values  of  #1  —  02  likely  to 
occur  in  practice,  let  us  suppose  that  6  =  1*649 
cm.,  and  a  =  l  cm.  Let  us  also  suppose  that  the  current 
density  i  is  150  amperes  per  sq.  cm.,  that  p=l'S  xlO~6 
and  that  k  =0-0006.  The  author  has  no  trustworthy  data 
with  reference  to  the  conductivities  of  the  dielectrics  used 
in  actual  cables,  and  so  he  takes  the  value  of  Tc  for  paraffin 
wax,  which  has  been  found  accurately  by  Lees  (I.e.  ante). 
Substituting  in  the  formula,  we  get 


0      e  _  (150)2  x  1-8  xlO-6     1 

8.4x0-0006 
=4°  C.  nearly. 

It  is  easy  to  see  from  the  formula  for  0±  —  02,  that  for  a 
given  value  of  b  and  for  a  given  current  density,  the  difference 
of  temperature  between  the  inner  and  outer  conductors  is  a 
maximum  when 

a  =  b/  v7  =  6/1-649  =  0-6065  6, 

which  is  the  case  we  considered.  We  see,  therefore,  that 
the  difference  of  temperature  between  the  inner  and  outer 
conductors  is  probably  not  greater  than  10  deg.  in  the  most 
unfavourable  circumstances. 


214        ELECTRIC  CABLES  AND  NETWORKS 

It  is  known  that  the  dielectric  coefficient  and 
Effects  of 

heat  on  the  the  electric  insulativity  of  an  insulating  material 
vary  rapidly  with  the  temperature.  Jona  (p.  207) 
mentions  a  case  where  a  rise  of  temperature  of  20°  C. 
made  the  insulation  resistance  of  a  paper  insulated  cable 
fall  to  one- thirtieth  of  its  original  value,  and  even  more 
striking  instances  could  be  given.  The  table,  also,  quoted 
in  the  last  chapter  (p.  196)  for  oven-dried  cellulose  shows 
that  the  dielectric  coefficient  varies  from  6-7  to  7 -5  as  the 
temperature  rises  from  20  to  70  degrees  Centigrade.  In  most 
cables  it  is  noticed  that  the  capacity  current  increases  and 
the  insulation  resistance  diminishes  as  the  temperature 
rises.  The  rise  of  the  capacity  current  is  probably  generally 
due  to  the  increase  in  the  value  of  the  dielectric  coefficient. 
We  see  therefore  that  in  practice,  since  the  temperature 
of  the  dielectric  is  not  uniform,  both  p  and  X  vary,  even 
when  the  insulating  covering  is  made  of  homogeneous 
material.  We  shall  now  consider  what  effect  this  has  on 
the  electric  stresses  in  the  insulating  material. 

Let  us  suppose  that  a  steady  pressure  E  is 

the          applied  across  the  inner  and  outer  conductors 
temperature       f  .    .  .      ,        .  .  ...  , 

gradient       of  a  concentric  mam  having  an  isotropic  dielec- 

eiectric       trie.     The  momentary  stresses  set  up  initially 


are  the  same  as  if  the  resistivity  were  infinite. 
Now  imagine  that  the  dielectric  is  split  up  into  an  infinite 
number  of  concentric  cylindrical  tubes,  the  material  of  each 
tube  being  at  the  same  temperature.  Since  these  tubes 
form  condensers  in  series  between  the  conductors,  the 
quantity  of  electricity  per  unit  length  in  each  condenser 
will  be  the  same,  and  thus 

X  -     -  dv  =  constant, 


where  X  is  the  value  of  the  dielectric  coefficient  at  a  distance 


THE   HEATING   OF  CABLES  215 

r  from  the  axis,  and  v  is  the  potential  at  the  same  distance. 
Hence 

_  dv  _  A 
~^dr~  \r9 

where  A  is  a  constant.  Now  X  diminishes  as  the  tempera- 
ture diminishes,  it  therefore  diminishes  as  r  increases.  We 
see,  therefore,  that  the  effect  of  A,  varying  with  the  tem- 
perature is  to  make  the  electric  stress  on  the  dielectric  more 
uniform. 

If  we  assume  that  X  varies  with  temperature  according 
to  the  linear  law,  we  may  write 


=  X0{l+£loge(6/r)}, 

where  B  =  a  a2  i2p  /&-4Jc.     It  readily  follows  that  the  electric 
stress  is  a  minimum  where 

r=be  l~ B)/B. 

In  practice,  B  is  very  small  compared  with  unity,  and 
hence  the  electric  stress  diminishes  as  we  pass  from  the 
inner  to  the  outer  conductor. 

Let  us  now  suppose  that  the  direct  pressure  E  has  been 
applied  sufficiently  long  to  make  the  electric  stresses  and 
the  leakage  currents  assume  their  steady  values.  In  this 
case,  by  Ohm's  law, 

duff  <r  —\=  constant, 


where  a  is  the  insulativity,  in  ohms,  of  the  dielectric. 

Hence  —dv-=—A\ 

cr  dr 

dv       a  A' 

or  —  —  =  -  - , 

dr         r 

where  A'  is  a  constant. 

Now    a   increases    as    the  temperature    diminishes    and 
therefore  as  r  increases.     The  variation  of  cr,  therefore,  due 


216        ELECTRIC  CABLES  AND  NETWORKS 

to  a  slight  temperature  gradient  in  the  dielectric  tends 
again  to  make  the  stress  more  uniform.  But  if  there  be  a 
drop  of  10°  C.  between  the  inner  and  outer  conductors  the 
electric  stresses  on  the  outer  layers  of  the  dielectric  when 
the  cable  is  loaded  may  be  much  greater  than  on  the  inner 
layers. 

The  The  variation  of  temperature  over  the  cross- 

temperature    section  of  a  conductor,  once  the  flow  of  heat  has 

in  tne 

ofSccmducfors    attained  its  steady  state,  is  very  small.     Let  us 

uniform       consider,  for  instance,  a  cylindrical  copper  con- 

currents      ductor  carrying  a  constant  current  the  density 

of  which  is  i,  so  that  the   total   current  is   i.irB*.     Then 

when  the  flow  of   heat   has  become   steady,  the   heat  per 

unit  length  flowing  across  the   surface  of  an   elementary 

concentric  cylinder  of  radius  r  must  equal  the  heat  being 

generated  per  unit  length  by  the  current  flowing  inside 

this  elementary  cylinder.     Thus  we  have 

k.27rr(d0/dr)  =  — 


Therefore  dO/dr  =  —  (Pi2/8-±k)r 

and  thus  6=  00+(Pi2/16-8k)(R*—  r2). 

If  &i  be  the  maximum  temperature  which  will  be  along  the 

axis  of  the  conductor,  we  have 


As  an  example  let  us  take  the  case  of  a  cylindrical  copper 
main,  the  diameter  of  which  is  2  cms.,  so  that  E  =  l.  We 
shall  suppose  that  /?=1'68  xlO""6  ohms,  £  =  100  amperes 
and  &=0»96  C.G.S.  units.  In  this  case  the  maximum  dif- 
ference BI  —  00  between  any  two  points  on  the  cross-section 
of  the  main  is  given  by 

0!—  00=l-68x  10~6x  10*7(16-8  xO-96) 

=  0-001°  C. 
Hence  no  appreciable  error  is  made  in  this  case  by  the 


THE   HEATING   OF  CABLES  217 

assumption  that  the  temperature  is  uniform  over  the  cross- 
section  of  the  core.  Even  if  we  suppose  that  the  resistivity 
of  the  metal  is  ten  times  that  of  copper  and  that  the  current 
density  is  1,000  amperes  per  square  centimetre,  the  maxi- 
mum difference  of  temperature  would  only  be  1°  C. 

The  thermal  conductance  of  the  dielectric  of 

The  thermal  -,  ,  ,      .      .,  f    ,-.        „  f 

conduc-       a  single-core  cable  is  the  ratio  of  the  now  of 

tance  of  a       -,  •*  JTTIJ-- 

single-core  heat  per  second  across  the  dielectric,  in  gramme- 
Centigrade  units,  to  the  difference  in  tempera- 
ture (Cent.)  between  the  outer  boundary  of  the  core  and 
the  inner  boundary  of  the  lead  sheath,  when  the  flow  of  heat 
has  attained  its  steady  state.  As  the  thermal  conductivities 
of  insulating  materials  are  very  small  compared  with  those 
of  metals,  we  can  assume,  without  appreciable  error,  that 
the  metals  are  at  uniform  temperatures.  If  a  and  b  be  the 
inner  and  outer  radii  of  the  dielectric  .(supposed  isotropic) 
we  have  (p.  212) 

B  =  0a+(01--02)  {log.  (b/r)/log£(b/a)  }  , 
and 

Q  =  —  k.27rr(dO/dr) 

A          n 
!  -  "2 


i  —  Tn-\> 
loge(&/a) 

where  B  is  the  temperature  of  the  dielectric  at  points  distant 
r  from  the  axis  of  the  cable,  B±  and  62  the  temperatures  of 
the  core  and  sheath  respectively,  and  Q  is  the  thermal  flow 
per  unit  length  in  calories  per  second.  Hence  the  thermal 
conductance  per  unit  length  for  a  single-core  cable  is 
27rk/loge(b/a)-.  It  is  therefore  equal  to  47T&/A,  times  the 
corresponding  electrostatic  capacity.  If  ri  be  the  electric 
resistance  of  the  conducting  core  per  unit  length  and  C  the 
current  flowing  in  it,  then,  when  the  thermal  flow  attains 
its  steady  value,  Q=C72ri/4«18,  and  thus  6±  —  Q2  can  be 
readily  found  if  k  be  known. 


218        ELECTRIC   CABLES   AND  NETWORKS 

Numerical         -^et  us  assur&e  that  the  diameter  of  the  con- 
examples      ductor  of   a  single-core  main  is   1    centimetre, 
and  that  a  current  of  100  amperes  is  flowing  in  it.     Let  us 
also  assume  that  6/a=2'718,  and  that  k  is  0-0004.     In  this 
case,  n  will  be  0-2xlO~5  ohms  approximately,  and   thus 
Q=£2rt/4-18  =0-02/4-18  =0-004785. 
Hence  since 


we  readily  find  that 

0!—  02  =  l-9°  C.  nearly. 

If  we  suppose  that  k  is  greater  than  0-0004  the  difference 
of  the  temperatures  will  be  less  than  this. 
The  thermal       We   shall   now   consider   a   concentric   main. 

conduc- 

tance  of  a     Let  the  conductivity  of  the  isotropic  dielectrics 
concentric 

main  between  the  two  conductors  and  between  the 
outer  conductor  and  the  lead  sheath  be  &t  and  k2  respectively. 
Then  if  0^  ,02,  and  63  be  the  temperatures  of  the  two  con- 
ductors and  the  sheath,  we  have 

Q/2=27rki(ei—  0a)/loge(6/a), 
and 


where  Q/2  is  the  heat  generated  per  unit  length  in  each 
conductor  per  second,  and  c  and  d  are  the  radii  of  the  outer 
dielectric.  Hence 

Q  27T 


If  we  can  write  b  =  c,  and  ki  =  k2,  without  appreciable  error, 
we  get  2rf1/loge(d/v/i&)  for  the  thermal  conductance. 

The  problem  of  the  thermal  conductance  of  a 

The  thermal 
conduc-       polycore  cable  is  much  more  difficult  than  that 

poiycore       of  the  single-core  cable.     When  the  dielectric  is 


isotropic,  formulae  for  the  electrostatic  capacity 
between  the  cores  in  parallel  and  the  sheath  are  given  in 


THE  HEATING  OP  CABLES 


219 


Russell's  Alternating  Cur- 
rents, vol.  i,  chap.  v. 
The  corresponding  ther- 
mal conductances  can 
be  at  once  deduced  from 
these  formulae  by  multi- 
plying the  capacities  by 
47T&/X.  In  a  three-core 
cable,  for  instance,  of  a 
certain  "  clover  leaf " 
pattern  (Fig.  55),  if  Q  be 
the  heat  generated  in 
the  three  cores,  per  unit 
length,  we  have 

Q  . 


FIG.  55. — The  section  of  a  Three-core 
Main  of  which  the  thermal  conduc- 
tance can  be  accurately  calculated. 


—  c3)]' 

exactly,  where  R  is  the  maximum  inner  radius  of  the  lead 
sheath   and  b  and  c  are  the  maximum  and  minimum  dis- 

tances of  points  on 
the  cores  from  the 
axis  of  the  cable. 

By  using  Kelvin's 
method  of  images,  it 
may  be  shown  that 
if  the  centres  of  the 
n  cores  are  symmetri- 
cally situated  on  a 
circle  of  radius  a 
(Fig.  56),  and  if  the 
cross-sections  of  the 

COr6S      ar6      Sma11    Cir" 


FIG.    56.—  Section  of  another   ideal  Three 

core  Main,  of  which  the  thermal  con-       cles    of 

ductance   can    be    calculated    approxi- 

mately. have, 


radlUS    T,    WG 


220        ELECTRIC   CABLES   AND   NETWORKS 
Q 


approximately,  where  Ris  the  inner  radius  of  the  lead  sheath. 
When  n  =  3  this  formula  agrees  with  that  given  above, 
provided  that  r/a  and  (a/E)3  can  be  neglected  compared 
with  unity. 

It  is  to  be  noticed  that  all  the  logarithms  given  in  this 
chapter  and  the  preceding  two  chapters  are  Neperian. 
Their  values  are  best  found  directly  from  Bottomley's 
Tables.  They  may  also  be  found  with  an  accuracy  sufficient 
for  practical  purposes  by  multiplying  the  corresponding 
ordinary  logarithms  by  2-3.  In  many  cases  they  may  be 
computed  easily,  without  tables,  by  means  of  the  formula 


where 

»={(&/<»)—  1}  {(&/»)+!}. 
For  instance 

log.  1-5=  0-4+0-016/3+  ... 
—  0-405. 

We  shall  conclude  this  chapter  by  discussing 
The  heating 
of  bare       the   variation   of   the   temperature    of   a    bare 

cylindrical  conductor,  through  which  electrical 
currents  are  passing,  when  suspended  horizontally.  We 
have  seen  that  no  appreciable  error  is  made  by  the  assump- 
tion that  the  temperature  of  the  conductor  is  uniform  over 
its  cross-section.  We  shall  make  the  further  assumption 
that  the  rate  at  which  the  wire  radiates  heat  is  proportional 
to  its  surface  and  to  the  difference  of  temperature  between 
the  wire  and  its  surroundings.  Hence  for  rises  of  tem- 
perature greater  than  about  50°  C.  our  results  are  only 
rough  approximations.  The  temperatures  of  the  conductor 
also  will  not  be  the  same  during  a  gale  or  when  it  is  raining 
as  they  would  on  a  calm  dry  day.  The  formulae,  however, 


THE   HEATING   OF   CABLES  221 

indicate  how  the  temperatures  vary  with  the  dimensions 
of  the  conductor. 

Let  us  first  suppose  that  when  the  switch  is  closed  a 
current  C  flows  in  the  conductor,  and  that  this  current  is 
maintained  constant.  Let  p  be  the  volume  resistivity, 
D  the  density,  h  the  emissivity  for  heat  of  the  surface, 
c  the  specific  heat,  /  the  length,  and  r  the  radius  of  the  con- 
ductor. If  6  be  the  temperature  of  the  wire,  at  the  time  t 
after  the  switch  has  been  closed,  we  have 


since  the  electric  power,  expressed  in  calories  per  second, 
being  expended  in  the  wire  equals  the  rate  at  which  heat 
is  being  radiated  from  the  surface  together  with  the  rate 
at  which  it  is  being  stored  in  the  substance  of  the  conductor. 
Assuming  for  the  present  that  p  is  constant,  we  see  that  the 
solution  of  the  above  equation  is 

0=^(1—6-^), 

where  0±  =  0  -239(7  2/?/(2/br2r3)  and  m=2h/Drc. 

The  final  temperature  of  the  wire  is  6±  which  varies  directly 

as  C2p  and  inversely  as  hr3. 

Let  us  next  suppose  that  the  voltage  V  at  the  terminals 
of  the  wire  is  maintained  constant.  The  equation  now 
becomes 


and  thus, 
where, 


Hence  the  final  temperature  varies  directly  as  (V2/p)  and  r, 
and  inversely  as  hi2.  The  greater  the  value  of  m  the  more 
rapidly  does  the  temperature  rise  to  its  steady  value.  The 
greater  the  value  of  the  emissivity,  therefore,  and  the 


222        ELECTRIC  CABLES   AND  NETWORKS 

smaller  the  value  of  the  density,  the  radius,  and  the  specific 
heat,  the  more  rapid  will  be  the  rise  of  temperature. 

Let  us  now  suppose  that  p  varies  with  the  temperature. 
In  this  case  we  may  assume  that  p  =  p0(I+a6)  where  a  is 
a  constant.  When  the  current  is  constant  the  temperature 
equation  now  becomes 


and  hence, 

0=041—  IT**), 

where  0t  =  0-239CV0/(2forar8—  0-239(7  zPoa), 

and  ra  =  (2hr-Q&39C*p0a/ir*r*)/Dre. 

Again  when  the  applied  voltage  is  constant,  noticing  that 
l/p=l/p0  —  aO/p0  approximately,  we  have 

0-239  V*(7rr*/Pol)  =  {  27rr^+0-239F2a(7rr2//>00  }  0 

+D7rr2lc(d0/dt), 

and  thus  6'  =  0-239  V2r/(2hl2p  0  +0-239  F2«r), 

and  m'  =  {2^-f  0-239F2a(r/^2)}  /Drc, 

and  the  temperature  at  any  instant  is  given  by 


REFERENCES. 

G.  Mie.     "  tiber  die  Warmeleitung  ineinem  verseilten  Kabel."    Elek- 

trotechnische  Zeitschrift,  vol.  26,  p.  137,  1905. 
R.  V.  Picou.     "  Capacite  et  BchaufTement  des  Cables  Souterrains." 

V  Industrie  Electrique,  vol.  15,  pp.  245  and  281,  1906. 
A.    Russell.     "  The   Dielectric    Strength   of   Insulating   Materials." 

Journ.  Inst.  EL  Engin.,  vol.  40,  p.  6,  1907. 
A.  E.  Kennelly.      "  The  Heating  of  Copper  Wires  by  Electric  Cur- 

rents."    Amer.  Inst.  Elect.  Engin.  Proc.,  vol.  26,  p.  39,  1907. 
G.  F.  C.  Searle.     "  A  Method  of  Determining  the  Thermal  Conduc- 

tivity of  'India  Rubber."     Camb.   Phil.   Soc.   Proc.,  vol.   xiv., 

Part  II,  1907. 


ELECTRICAL    SAFETY    VALVES 


CHAPTER    XI 

Electrical   Safety   Valves 

Electrical  safety  valves — Intermittent  safety  valves — Siemens  and 
Halske  horn  arrester — The  Seibt  safety  valve — Multiple  gap 
lightning  arresters — Pressure  safety  valves  on  a  3-phase  line — 
Continuous  arresters — Electrolytic  arresters — References. 

IN  practical  working,  the  machines,  apparatus, 
Electrical 
safety        and   cables,  used   for   high  pressure    networks 

are  often  subjected  to  abnormal  stresses  owing 
to  a  sudden  rise  of  pressure.  This  rise  of  pressure  may  be 
due  to  atmospheric  electricity,  to  resonance  due  to  a  har- 
monic of  the  applied  E.M.F.  wave  having  the  same  period 
as  a  free  period  of  vibration  of  the  system,  or  to  electro- 
mechanical resonance  between  the  prime  mover  and  the 
oscillating  electrical  energy.  It  may  also  be  due  to  resonance 
of  the  high  frequency  oscillations  often  set  up  when  an 
arc  occurs  at  a  short  circuit.  Hence  excessive  rises  of 
pressure  can  occur  on  a  direct  current  network  when  an 
arc  giving  rise  to  Duddell  currents  exists  on  any  part  of  the 
circuit.  The  impulsive  rush  of  electricity  also,  at  these 
high  frequencies,  often  causes  momentary  disruptive  dis- 
charges, making  pinhole  marks,  through  the  dielectric 
at  places  where  the  rush  meets  with  inductive  resistance. 
Between  the  first  two  turns,  for  instance,  of  a  coil  in  the 
circuit  or  at  a  sudden  bend  of  the  conductor.  The  most 
frequent  cause  of  breakdown  is  due  to  the  oscillating  arc, 

225  Q 


226        ELECTRIC  CABLES   AND   NETWORKS 

but  the  most  disastrous  occur  when  the  period  of  one  of 
the  free  oscillations  of  a  network  has  the  same  value  as  a 
harmonic  of  low  order  of  the  impressed  oscillations.  In 
this  case  the  amplitude  of  the  pressure  between  two  points 
may  reach  enormous  values,  and  considerable  power  may 
be  expended  where  a  breakdown  occurs. 

To  prevent  the  breakdown  of  the  cables  from  these 
causes,  or  of  the  insulation  of  the  armatures  of  the  dynamos 
and  other  appliances  in  the  network,  electrical  "  safety 
valves  "  must  be  provided.  These  devices  may  be  classed 
under  two  main  heads,  (1)  intermittent  safety  valves, 
that  is,  devices  which  only  act  when  the  pressure  exceeds  a 
certain  critical  value,  and  (2)  continuous  safety  valves  which 
are  always  in  operation.  The  first  type  acts  by  providing  a 
safety  path  for  the  oscillating  charge  when  its  value  gets 
excessive.  The  second  type  acts  by  conduction.  It  pre- 
vents the  accumulation  of  an  excessive  charge  by  allowing 
it  to  leak  away  by  a  path  of  small  resistance  which  is  always 
in  circuit. 

In    the    usual    types  of   intermittent    safety 

Inter-        valves,  the  two  electrodes  are  separated  by  a 
mittent 

safety        suitable  dielectric  which  is   usually  either    air 
valves 

or  oil.  One  electrode  is  connected  with  one 
main,  and  the  other  with  another  main  of  different  polarity. 
When  the  difference  of  the  pressure  between  the  two  ex- 
ceeds a  certain  value  the  dielectric  breaks  down,  and  the 
ensuing  arc  having  a  very  small  resistance,  the  pressure 
between  the  mains  to  which  the  device  is  joined  cannot 
attain  a  high  value.  The  ensuing  arc  is  broken  in  several 
ways,  some  of  which  are  described  below. 

If  one  of  the  electrodes  is  connected  with  a  main,  and  the 
other  with  an  earth  plate,  the  device  is  generally  called  a 
lightning  arrester.  A  device  used  to  limit  the  pressure 


ELECTRICAL  SAFETY  VALVES 


227 


Siemens 

andhoraSke 
arrester 


©  M 


can  also  be  used  as  a  lightning  arrester.  The  breadth  of  the 
gap  between  the  electrodes,  and  the  design  of  certain  auxiliary 
apparatus,  however,  is  generally  different  in  the  two  cases. 
The  Siemens  and  Halske  horn  arrester 
57)  is  perhaps  the  one  most  extensively 
used  on  power  circuits.  When  the  pressure 
between  the  main  M  and  the  earth  E  exceeds  a  certain 
value,  an  arc  ensues  between  the  narrowest  portion  of  the 
gap  between  the  two 
horns.  It  then  rapidly 
travels  upwards  until 
its  length  gets  too 
great  for  the  voltage 
at  its  terminals  when 
it  automatically  rup- 
tures. Although  the 
rupture  of  the  arc  is 
accelerated  by  the 
convection  currents 
of  air  yet  it  is  partly 
also  an  electromag- 
netic phenomenon. 
If  we  invert  the  ar- 
rester, for  instance, 
the  arc  travels  downwards  if  it  is  started  below  the  nar- 
rowest point  (see  A.  Moore,  Elect.  Engineer,  vol.  34.,  p.  520, 
1904). 

The  Oerlikon  Company  use  a  horn  arrester  with  a  non- 
inductive  resistance  in  series  with  it.  This  is  formed  of 
nickeline  wire  immersed  in  oil  and  bent  so  as  to  be  practic- 
ally non-inductive.  The  use  of  this  resistance  is  to  obviate 
the  dangers  arising  from  possible  high  frequency  oscillations 
being  set  up  at  the  arc,  and  also  from  oscillations  being  set 


FIG.  57. — Horn  Lightning  Arrester. 


228        ELECTRIC  CABLES   AND   NETWORKS 

up  by  a  sudden  rush  of  current  through  a  path  of  small 
resistance.  The  Allgemeine  Elektricitats  Gesellschaft  have 
employed  for  the  series  resistances  150  watt,  150  volt  glow 
lamps  connected  in  series,  a  sufficient  number  being  em- 
ployed to  prevent  any  of  them  burning  out  and  breaking 
the  circuit. 

To  prevent  oscillations,  and  to  provide  a  path  of  small 
resistance  for  the  discharge,  it  is  important  to  have  the 
inductance  of  these  circuits  as  low  as  possible.  In  practice, 
it  is  difficult  to  make  the  inductance  of  a  wire  circuit  small 
enough,  and  hence  carbon  cylinders,  water  resistances,  and 
even  wet  sand  are  sometimes  employed.  The  function  of 
these  resistances  is  to  get  rid  of  most  of  the  destructive 
energy  contained  in  violent  atmospheric  discharges. 

W.  H.  Patchell  (Journ.  Inst.  EL  Engin.,  vol.  36,  p.  97) 
has  found  that  a  spark-gap,  with  one  electrode  of  copper 
and  the  other  of  carbon,  in  a  glass  enclosure  is  very  effective 
as  a  safety-valve.  The  travelling  of  the  spark  upwards  in 
the  gap  between  the  horns  is  accelerated  by  the  chimney 
action  of  the  enclosure,  and  many  tests  have  proved  that 
this  type  can  be  calibrated  more  accurately,  and  adjusted 
within  narrower  limits,  than  the  ordinary  open  horn  type 
with  copper  electrodes.  A  liquid  resistance  is  used  con- 
sisting of  a  solution  of  glycerine  and  water  contained  in 
earthenware  vessels,  but  it  has  not  proved  altogether  satis- 
factory as  the  values  of  the  resistances  are  liable  to  change. 
The  width  of  spark-gap  employed  is  4-5  mms.  for  10,000^3" 
that  is,  about  5,800  volt  working.  A  spark  will  jump  the 
gap  and  start  the  arc  at  a  pressure  of  12,000  effective  volts 
when  the  horns  are  clean  and  the  atmosphere  is  normal. 

When  these  safety-valves  were  first  installed  in  the 
City  of  London  Works  of  the  Charing  Cross  Company,  the 
irregular  times  at  which  they  acted  attracted  attention. 


ELECTRICAL  SAFETY  VALVES 


229 


To  discover  the  cause  a  detector  was  extemporized  for  ex- 
perimental use.  The  primary  of  a  small  transformer  was 
inserted  in  the  earth  wire  of  the  spark-gap  resistance.  The 
secondary  acted  a  relay  which  rang  a  bell  and  thus  attracted 
the  attention  of  the  engineer.  It  was  found  that  irregu- 
larities in  starting  a  machine,  although  not  sufficient  to 
prevent  easy  synchronizing  and  switching  in,  were  a  frequent 
cause  of  spark  discharges.  An  interruption  in  the  supply 
due  to  a  faulty  insulator  nearly  always  caused  a  discharge. 
It  is  probable,  therefore,  that  the  rises  of  pressure  in  the 


FIG.   58. — Seibt  Lightning  Arrester. 

network  were  due  to  the  superposition  of  the  free  oscilla- 
tions set  up  by  the  disturbance  on  the  normal  oscillations. 
The  indicating  device  has  proved  so  useful  that  it  has  been 
adopted  permanently.  A  time  recorder  could  also  easily 
be  actuated  by  the  transformer,  the  inductance  of  the 
primary  of  which  can  be  made  very  small. 

A  drawback  to  the  use  of  the  horn  arrester 
The  Seibt 

safety        is  the    large    factor  of  safety  that  has  to   be 
valve 

allowed  in  order  to  avoid  unnecessary  sparking. 


230        ELECTRIC  CABLES   AND  NETWORKS 

Hence  it  might  easily  happen  that  the  excessive  electric 
stress  did  serious  harm  before  the  valve  acted.  This 
difficulty  is  neatly  surmounted  in  the  Seibt  safety  valve 
(Fig.  58),  by  utilizing  the  well-known  effect  of  ultra-violet 
radiation  in  lowering  the  value  of  the  disruptive  voltage 
required  by  an  air  gap.  The  primary  P  of  a  small  trans- 
former is  put  in  series  with  the  line  L,  and  the  secondary  8 
consists  of  many  turns  of  fine  wire.  A  vacuum  tube  T 
placed  between  the  secondary  terminals  glows  when  high 
frequency  oscillations  are  set  up  in  the  main,  and  the  di- 
electric strength  of  the  air  in  the  spark-gap  safety  valve 
being  lowered  by  the  radiations  from  the  vacuum  tube,  a 
disruptive  charge  takes  place  to  the  earth  E,  and  thus  the 
pressure  is  prevented  from  becoming  excessive. 

The  multiple  gap  lightning  arrester  (Fig.  59) 

Multiple  . 

gap  lightning   is  a  type  of  arrester  frequently  used  in  power 
arresters  . 

transmission  circuits  in  America.     Between  the 

line  M  and  the  earth  E  there  is  a  series  of  insulated  con- 
Line          M  Series  Air  Gctps  P  Shunted  Air  G<tps. 

oooooooooOoooopoooooooooooooAWV 

Res 

1 V\A/WWVN ' 

Res. 

FIG.  59. — Multiple  Gap  Lightning  Arrester. 

ductors,  having  small  air  gaps  between  them,  and  there  is 
a  resistance  in  series  with  the  last  of  the  conductors.  In 
order  to  prevent  an  excessive  rush  of  current  from  the  line 
when  the  device  acts,  it  is  necessary  to  make  this  resistance 
of  appreciable  magnitude  which  is  an  objectionable  feature 
as  it  lowers  the  efficiency  of  the  device.  To  obviate  this 
difficulty  another  resistance  is  placed  as  a  shunt  to  half 


ELECTRICAL  SAFETY  VALVES      231 

the  conductors  (Fig.  59),  and  it  is  consequently  in  series 
with  the  first  resistance.  Since  P  is  at  earth  potential 
initially,  a  discharge  ensues  when  the  voltage  is  sufficiently 
high  to  break  down  the  series  air  gaps  from  M  to  P.  The 
impulsive  rush  of  current  which  ensues  breaks  down  the 
shunted  air  gaps  and  gets  to  earth  by  the  series  resistance. 
When  the  impulsive  rush  is  over  the  arcs  across  the  shunted 
air  gaps  go  out,  being  shunted  by  a  resistance.  Both  re- 
sistances in  series  now  carry  the  current  through  the  arcs 
between  the  series  air  gaps,  and  hence,  when  the  voltage 
becomes  normal,  these  arcs  go  out  and  the  leakage  of  current 
from  the  main  is  stopped.  It  will  be  seen  that  the  use  of 
a  resistance  shunting  some  of  the  air  gaps  makes  the  device 
much  more  effective. 

The    Societe    d'energie    electrique   de    Grenoble 
Pressure 

safety       et    Voiron   have   found   the   following   arrange- 
valves 
on  a        ment  of  lightning  arresters  and  pressure  safety 

phase        valves   very   satisfactory  in   practical   working 
line 

for  an  overhead  15,000  volt  network,  extending 
over  60  kilometres,  in  a  district  subject  to  severe  thunder- 
storms. In  Fig.  60,  three  horn  lightning  arresters,  outside 
the  power  station  S,  one  connected  with  each  main,  are 
represented  at  A.  The  minimum  width  of  the  air  gap  is 
13  mms.,  and  each  earth  circuit  is  composed  of  damp  sand 
the  resistance  of  which  is  about  8,000  ohms.  No  choking 
coils  are  placed  between  the  arresters  and  the  mains,  but 
the  inductance  of  the  connecting  wires  is  increased  by 
bending  them  at  sharp  angles.  The  safety  valve  P,  limiting 
the  pressure  between  the  mains,  is  in  the  station  itself,  and 
consists  of  three  Siemens  horn  arresters,  connected  in  mesh, 
and  having  resistances  of  15,000  ohms  in  circuit  with 
them. 

Originally  lightning  arresters  were  placed  at  distances 


232        ELECTRIC  CABLES   AND  NETWORKS 

of  2-5  kilometres  apart  all  along  the  line.  These  are  now 
replaced  by  two  lightning  arresters  the  positions  of  which 
have  been  carefully  chosen.  These  have  been  found  quite 
sufficient  to  protect  the  line  insulators  from  damage  during 


I [ 


a 


FIG.   60. — Lightning  Arrester  A  and  device  P  for  limiting  the  pressure. 


thunderstorms.  Interruptions  to  the  working  of  the 
line  due  to  sudden  falls  of  the  potential  difference  and 
short  circuits,  which  were  formerly  often  caused  by  the 
irregular  action  of  the  earlier  types  of  arrester  used,  now 
practically  never  occur. 

At  the  transformer  substations  either  horn  or  multiple- 
gap  lightning  arresters  are  used.  No  other  special  safety 
valve  apparatus  is  employed  to  limit  the  rise  of  pressure 
between  the  mains  at  the  substations. 

Continuous         ^n  ^s  type  of  arrester  resistances  through 
arresters       which  a  current  is  continually  leaking  are  inter- 
polated between  the  lines  and  earth.     Hence  it  is  essential 
that  during  normal  working  the  losses  due  to  these  arresters 
should  not  be  large.     A  jet  of  water  is  generally  employed 


ELECTRICAL  SAFETY  VALVES 


233 


in  continuous  arresters  to  carry  the  leakage  current,  as  a 
comparatively  large  amount  of  energy  can  be  got  rid  of  in 
this  way  without  using  costly  resistances,  and  there  is  no 
risk  from  over  heating. 


-M 
-M 
-M 


FIG.  61. — Continuous  Arrester. 


The  Societe  hydro-electrique  de  Vizille  distribute  power, 
by  a  3 -phase  overhead  network  40  kilometres  long,  at  a 
pressure  of  10,000  volts.  As  this  pressure  is  obtained 
directly  from  the  terminals  of  the  machines,  special  pre- 
cautions have  to  be  taken  to  protect  the  armatures  of  the 
machines  from  damage  by  sparks  due  to  atmospheric  dis- 


234       ELECTRIC  CABLES  AND  NETWOBKS 

charges  causing  short  circuits  or  damaging  the  insulation. 
When  the  network  was  first  installed  only  rough  types  of 
arresters  were  used,  and  as  thunderstorms  are  frequent 
and  severe  in  this  district  the  3-phase  machines  were  often 
damaged. 

At  the  power  station  a  continuous  arrester  (Fig.  61)  is 
employed  and,  in  addition,  choking  coils  are  placed  in 
series  with  the  mains  to  hinder  impulsive  rushes  of  electricity 
from  getting  to  the  terminals  of  the  machines,  and  thus  into 
the  armature .  At  the  transformer  su bstations,  horn  arresters 
with  resistances  in  series  with  the  earth  connexion  are  used. 
The  resistances  are  simply  stoneware  tubes,  80  centimetres 
long,  filled  with  water. 

The  continuous  arrester  (Fig.  61)  consists  of  three  stone- 
ware tubes  each  of  which  is  2-5  metres  long  and  15  centi- 
metres in  diameter.  They  are  fixed  in  an  iron  pipe  5  metres 
long  and  20  centimetres  in  diameter  which  is  in  connexion 
with  a  good  earth.  A  current  of  water  is  continually  flowing 
up  the  stoneware  tubes  and  escaping  from  the  waste  pipes 
near  the  top.  Each  of  the  three  mains  is  in  direct  contact 
with  the  water  through  a  wire  which  dips  into  it  to  a  depth 
of  a  few  centimetres.  The  current  in  each  wire  during 
normal  working  is  about  0-3  of  an  ampere  and  thus  the 
power  expended  in  this  device  is  V3  x  10000 x  0-3  watts, 
that  is,  about  5  kilowatts. 

This  continuous  pressure  arrester  has  been  in  use  for  some 
years,  and  no  accidents  to  the  alternators,  due  to  atmo- 
spheric electricity,  which  were  formerly  frequent,  now  occur. 

Another  type  of  continuous  arrester  due  to  La  Societe 
d?  Applications  Industridles  is  shown  in  Fig.  62.  The  mains 
are  connected  with  earth  by  means  of  vertical  jets  of  water 
which  play  against  metallic  cups,  each  cup  being  in  direct 
connexion  with  a  main  through  a  wire.  Particular  care 


ELECTRICAL  SAFETY  VALVES 


235 


has  to  be  taken  that  the  pipe  bringing  the  water  has  a 
good  earth  connexion. 


I 


Electro- 
lytic 

arresters 


FIG.  G2. — Water  Jet  Continuous  Arrester. 

f 

Various  types  of  electrolytic  cell  possess  uni- 
lateral conductivity,  that  is,  they  allow  the 
electric  current  to  pass  through  them  much 
more  readily  when  it  flows  in  one  direction  than  when  it 
flows  in  the  other.  An  electrolytic  cell  may  be  made  by 
immersing  one  electrode  of  aluminium,  and  one  of  some 
other  conducting  substance,  in  an  electrolyte.  Dilute  sul- 
phuric acid,  bichromate  solution,  ammonium  phosphate 


236        ELECTRIC  CABLES   AND  NETWORKS 

solution,  etc.,  are  suitable  electrolytes.  If  the  aluminium 
electrode  be  at  a  higher  potential  than  the  other,  and  if  the 
P.D.  between  them  be  less  than  a  certain  critical  value, 
very  little  current  will  flow  through  the  cell.  This  is  owing 
to  the  formation  of  a  thin  film  of  high-resistance  material 
round  the  aluminium  electrode.  K.  Norden  (Electrician, 
vol.  xlviii,  p.  107)  has  found  that  this  film  consists  of  normal 
aluminium  hydroxide  [A12(OH)6].  If  the  direction  of  the 
applied  voltage  be  reversed,  the  film  dissolves  rapidly,  and 
the  effective  resistance  of  the  cell  is  very  considerably 
reduced. 

When  the  electrolytic  cell  is  placed  in  an  alternating 
current  circuit,  and  the  maximum  value  of  the  applied  P.D. 
is  less  than  the  critical  voltage  for  the  cell,  the  current 
flowing  through  it  during  the  half  period  when  the  aluminium 
electrode  is  at  the  lower  potential  will  be  much  greater  than 
during  the  other  half  period  and  hence  we  get  what  is 
practically  a  pulsating  direct  current.  This  is  the  principle 
utilized  in  the  Pollak  electrolytic  rectifier,  and  in  the  Nodon 
valve.  Both  of  these  rectifiers  can  be  employed,  for 
example,  for  charging  direct  current  accumulators  from  the 
alternating  current  mains. 

If  we  gradually  raise  the  direct  voltage  applied  to  the 
terminals  of  an  electrolytic  cell,  the  aluminium  electrode 
of  which  is  connected  with  the  positive  main,  then,  when 
the  critical  voltage  is  passed  the  current  increases  and  the 
resistance  of  the  cell  diminishes  very  rapidly.  It  is  this 
action  of  the  cell  that  makes  it  valuable  as  an  electrical 
safety  valve  for  preventing  pressure  rises  on  power  trans- 
mission lines  due,  for  instance,  to  a  change  in  the  normal 
working  of  the  system  or  to  an  impulsive  rush  of  electricity 
caused  by  a  disturbance  of  the  atmospheric  potential. 

If  both  the  electrodes  are  of  aluminium,  then,  at  an  altern- 


ELECTRICAL   SAFETY  VALVES  237 

ating  pressure  the  maximum  value  of  which  does  not  exceed 
the  critical  pressure,  very  little  current  will  pass  through  the 
cell,  and  at  a  pressure  the  maximum  value  of  which  is,  let 
us  suppose,  10  per  cent,  greater  than  the  critical  pressure 
a  very  large  current  will  pass.  Hence  a  battery  of  cells  of 
this  type  would  form  a  suitable  safety  valve  to  be  con- 
nected between  two  alternating  current  mains  to  prevent 
the  pressure  from  ever  becoming  excessive.  C.  Garrard 
(Electrician,  vol.  lix,  p.  147)  states  that  the  critical  voltage 
for  a  cell  having  two  aluminium  electrodes  dipping  into  a 
bichromate  solution  is  about  110  volts.  Hence,  for  a  safety 
valve  between  20,000  volt  alternating  current  mains,  about 
280  of  these  cells  would  be  required,  if  the  voltage  is  sine 
shaped  so  that  the  normal  maximum  pressure  is  28,280 
volts. 

One  effect  in  connexion  with  these  cells  which  has  to  be 
remembered  when  they  are  used  on  alternating  current 
circuits  is  that  they  act  as  electrostatic  condensers.  The 
thickness  of  the  film  round  the  aluminium  anodes  is  micro- 
scopic and  its  resistance  is  very  high.  The  P.D.  across  this 
film  is  appreciable,  and  thus  the  electrostatic  charge  due  to 
the  condenser  action  is  also  appreciable.  In  practice,  n 
of  these  condensers  are  connected  in  series,  and  thus  the 
resultant  capacity  of  the  battery  of  cells  between  the  mains 
is  only  the  nth  part  of  the  capacity  of  one  cell.  Although 
this  capacity  is  in  general  very  small  yet  with  the  very 
high  frequency  "  pressure  rises  "  sometimes  set  up  by  an 
arc  in  the  circuit,  the  condenser  current  may  appreciably 
relieve  the  pressure. 

The  electrolytic  lightning  arrester  is  generally  used  in 
conjunction  with  a  spark  gap  (Fig.  63).  As  there  is  no 
leakage  current  in  this  case,  there  is  no  risk  of  the  electrolyte 
evaporating  or  of  the  electrodes  being  deteriorated  by  over- 


238        ELECTRIC  CABLES   AND  NETWORKS 

heating.  The  air-gap  also  can  be  adjusted  to  act  within 
much  narrower  limits  than  when  an  ordinary  resistance  is 
used.  In  Fig.  63,  C  represents  a  pile  of  electrolytic  elements 
which  in  practice  would  be  enclosed  in  an  earthenware  pipe. 


FIG.  63.— Electrolytic  Arrester. 

Each  element  consists  of  a  shallow  aluminium  dish.  They 
are  separated  from  one  another  by  pieces  of  insulating 
material.  A  solution  of  bichromate  of  potash  is  sometimes 
used  for  the  electrolyte.  It  is  poured  in  the  top  dish  slowly, 
filling  it,  and  then  trickling  down  and  filling  all  the  other 
dishes  in  turn.  A  drop  of  transformer  oil  in  each  of  the 
trays  forms  a  thin  film  over  the  surface  of  the  electrolyte 
and  thus  hinders  evaporation.  When  the  critical  voltage 
across  a  battery  of  this  type  is  exceeded,  the  surfaces  of  each 
dish  are  covered  with  tiny  sparks  and  brush  discharges, 
indicating  the  points  where  the  insulation  resistance  of  the 
non-conducting  film  has  broken  or  is  breaking  down. 
In  a  type  of  lightning  arrester  described  by  R.  Jackson 


ELECTRICAL   SAFETY   VALVES  239 

(The  Electric  Journal,  vol.  iv,  p.  469)  very  similar  to  that 
described  above  (Fig.  63),  the  voltage  required  per  cell  is 
stated  to  be  400.  Hence  if  V  be  the  effective  voltage  of  the 
alternating  current  between  the  line  and  earth,  and  k  the 
amplitude  factor,  so  that  Vk  is  the  maximum  value  of  the 
voltage  the  number  of  cells  required  would  be  slightly  more 
than  F&/400.  If  the  voltage,  for  instance,  between  the 
line  and  earth  is  10,000,  and  the  pressure  wave  is  sine  shaped, 
the  number  of  cells  required  would  be  slightly  more  than 
10,000x1-414/400.  Hence  40  would  be  sufficient. 

REFERENCES. 

P.  H.  Thomas,  "  The  Function  of  Shunt  and  Series  Resistance  in 
Lightning  Arresters."  Amer.  Inst.  Elect.  Engin.  Proc.,  19, 
p.  1021,  1902. 

Dusaugey,  "  Methode  de  Protection  centre  les  Surtensions  actuelle- 
ment  employee  dans  les  Reseaux  de  Transport  d'Energie."  Soc. 
Int.  Elect.  Bull,  vol.  5,  p.  109,  1905. 

P.  H.  Thomas,  "  An  Experimental  Study  of  the  Rise  of  Potential  on 
Commercial  Transmission  Lines  due  to  Static  Disturbances 
caused  by  Switching,  Grounding,  etc."  Amer.  Inst.  Elect.  Engin. 
Proc.,  24,  p.  705,  1905. 

G.  Seibt,  "  tJber  Spannungserhohungen  in  elektrischen  Leitungen 
und  Apparaten."  Elektrotechnische  Zeitschrift,  vol.  26,  p.  25, 
1905. 

P.  H.  Thomas,  "  Practical  Testing  of  Commercial  Lightning 
Arresters."  Amer.  Inst.  Elect.  Eng.  Proc.,  vol.  26,  p.  915,  1907. 
See  also  the  Bibliography  given  in  Appendix  3  of  this  paper. 

C.  C.  Garrard,  "  The  Electrolytic  Lightning  Arrester."  The  Elec- 
trician, vol.  58,  p.  858  and  vol.  59,  p.  147,  1907. 

R.  P.  Jackson,  "  The  Electrolytic  Lightning  Arrester."  The  Elec- 
tric Journal,  vol.  iv,  p.  469,  1907. 

"  The  Schneider  Water  Jet  Lightning  Arrester."  Electrical  World, 
vol.  1,  p.  767,  1907. 

J.  S.  Peck,  "  Protective  Devices  for  High-Tension  Transmission 
Circuits."  Journ.  of  the  Inst.  of  Elect.  Engin.,  vol.  40,  p.  498, 1908. 


LIGHTNING    CONDUCTORS 


CHAPTER  XII 

Lightning  Conductors 

Lightning  conductors — Atmospheric  electricity — Potential  gradient 
of  the  atmosphere— Kew  observations  —  Thunderstorm 
potential  gradients — The  potential  difference  required  for  a 
lightning  flash— The  A  flash— The  B  flash— Lightning  rods— 
The  current  in  the  conductor — Numerical  example — Side  flash 
— Lightning  Research  Committee— Earthing— Tubular  earth— 
The  metal  of  the  conductor — The  elevation  rod — Town  houses — 
Lightning  fatalities — References. 

Lightnin  ^s  lightning  conductors  for  protecting  buildings 
conductors  from  lightning  have  now  been  in  use  for  over  a 
hundred  years,  there  is  naturally  plenty  of  information  avail- 
able to  illustrate  the  effects  produced  when  a  lightning  flash 
strikes  a  conductor.  It  is  only,  however,  comparatively 
recently,  mainly  owing  to  the  researches  of  Sir  Oliver 
Lodge,  that  a  satisfactory  theory  has  been  developed  to 
explain  the  phenomena.  We  shall  first  briefly  consider  the 
causes  of  thunderstorms,  and  then  discuss  in  detail  the 
function  of  lightning  conductors,  or  as  they  are  frequently 
called,  lightning  rods. 

Atmospheric  -^  *s  universally  admitted  that  a  lightning  flash 
electricity  -g  a  phenomenon  similar  to  that  which  ensues 
when  a  Leyden  jar  is  discharged  by  a  spark.  It  is  clearly 
due  to  electricity  in  the  air.  During  thunderstorms  char- 
acteristic black  clouds  are  observed,  and  the  flash  takes 
place  between  a  cloud  and  the  earth  or  between  two 

243 


244        ELECTRIC  CABLES   AND  NETWORKS 

clouds.  We  may  conclude  that  previous  to  the  flash  a 
high  potential  difference  must  exist  between  the  two 
conductors  subsequently  short  circuited  by  the  flash. 

We  have  next  to  consider  what  produces  this  potential 
difference  between  a  stratum  of  the  atmosphere  and  the  earth 
or  between  two  different  atmospheric  strata.  The  friction 
between  neighbouring  strata  moving  with  different  velo- 
cities probably  developes  static  charges  on  the  minute  drops 
of  water  carried  along  by  the  air  currents.  Any  alteration 
of  the  level  of  the  strata  will  rapidly  alter  the  potentials  of 
these  charges.  As  the  dielectric  strength  of  air  is  not  very 
great,  the  electric  stress  will  sometimes  cause  a  disruptive 
discharge,  which  may  travel  considerable  distances  owing  to 
the  violent  equalizations  of  the  potentials  and  consequent 
increase  in  the  electric  stresses  between  other  strata  which 
may  ensue  as  the  flash  proceeds  almost  instantaneously 
from  one  stratum  to  another.  The  energy  originally  ex- 
pended, by  the  sun's  heat  in  vaporizing  and  raising  water  to 
heights  in  the  air,  is  converted  during  the  flash  into  heat,  light, 
sound,  and  electric  waves  radiating  into  space. 
Potential  ^e  results  obtained  by  many  experiments 
gof  ^he*  prove  that  there  is  practically  always  a  difference 
atmosphere  of  potential  between  atmospheric  strata  which 
are  at  different  heights,  the  positive  potential  normally 
increasing  with  the  height.  From  the  results  obtained  in 
numerous  balloon  ascents  F.  Linke  (Meteorologische  Zeitschrift, 
vol.  22,  p.  237)  finds  that  if  V  be  the  potential  in  volts,  and 
h  the  height  in  metres  above  the  ground,  then,  from  1,500 
to  6,000  metres  (his  highest  observation) 
dF/«=34— 0-006/L 

=  25—0-006(^—1,500). 

The  potential  gradient  of  the  atmosphere  dV/dh,  therefore, 
diminishes  the  farther  we  get  away  from  the  earth.  Up  to 


LIGHTNING  CONDUCTORS  215 

1,500  metres  he  finds  that  the  gradient  varies  from  day  to 
day,  but  above  this  height  the  gradient  seems  to  be  prac- 
tically the  same  over  long  periods.  If  we  accept  the  above 
formula,  the  potential  at  4,000  metres  is  nearly  44,000  volts 
above  that  at  1,500  metres.  If  we  assume  for  the  average 
gradient  up  to  1,500  metres,  the  mean  of  that  at  the  ground 
in  Linke's  experiments,  namely  125  v/m,  and  that  at  1,500 
metres  (25  v/m),  we  find  that  the  potential  at  1,500  metres 
equals  75x1,500,  that  is,  110,000  volts  approximately. 
This  gives  for  the  potential  difference  between  a  stratum  of 
air  4,000  metres  high  and  the  earth  about  150,000  volts. 

A  potential  gradient  of  125  volts  per  metre,  except  during 
midsummer,  is  really  a  low  ground  value  to  assume.  In 
winter,  if  the  weather  be  fine  its  value  is  generally  about 
300  volts  per  metre,  and  in  foggy  weather  it  is  sometimes 
1,000  volts  per  metre.  Thus  the  above  estimate  is  probably 
often  much  exceeded. 

C.   Chree   by  analysing  the  readings  of  the 

observa-  Kelvin  water- dropping  electrograph  at  Kew 
Observatory  has  found  that  there  are  two  dis- 
tinct daily  maxima  and  minima  values  of  the  potential 
gradient.  In  all  months  the  minima  occur  near  4  a.m.  and 
2  p.m.  The  times  at  which  the  maxima  occur  are  more 
variable.  He  also  finds  that  the  day  interval  between  the 
forenoon  and  the  evening  maximum  is  longer  in  summer 
than  in  winter. 

The  month  showing  the  highest  mean  potential  gradient 
is  December,  but  the  amplitude  of  the  diurnal  inequality 
is  greatest  in  February.  With  the  exception  of  the  month 
of  July  a  high  mean  potential  and  a  large  diurnal  range  of 
potential  were  found  associated  with  a  low  temperature. 

It  has  to  be  remembered  that  these  results  are  strictly 
applicable  to  Kew  only.  It  is  highly  probable  that  in 


246        ELECTRIC  CABLES  AND  NETWORKS 

mountainous  districts  the  electrical  atmospheric  phenomena 
would  be  different.  The  tendency,  in  winter,  to  a  single 
diurnal  period  visible  at  Kew  is  more  pronounced  elsewhere. 

It  is  interesting  to  remember  the  importance  that  Kelvin 
attached  to  a  study  of  atmospheric  electricity.  The  Kew 
water-dropping  electrograph  which  Chree  used  in  his  obser- 
vations was  probably  the  first  one  ever  made.  Kelvin  came 
to  Kew  and  had  it  put  up  under  his  immediate  supervision. 

Thunder-  ^e  va-lues  of  the  potential  differences  during 

potential      a  thunderstorm  have  not  yet  been  measured, 

gradients  ^u^  ^he  potential  gradients  near  the  ground 
are  sometimes  at  least  ten  times  greater  than  on  ordinary 
days.  In  mountainous  districts,  generally  when  the  air  is 
dry,  but  sometimes  even  during  rain,  brush  discharges  occa- 
sionally take  place  from  pointed  objects  showing  that  the 
potential  gradient  is  very  high.  The  action  is  sometimes  so 
energetic  that  a  hissing  noise  is  heard. 

The  black  appearance  of  a  thundercloud  may  be  easily 
imitated  by  putting  a  point  maintained  at  a  high  potential 
into  the  steam  escaping  from  a  kettle.  The  effect  of  elec- 
trifying a  drop  of  water  is  to  diminish  the  value  of  the  hydro- 
static surface  tension  and  hence  the  electrified  globules  of 
steam  coalesce  giving  a  much  darker  shade  to  the  cloud  of 
escaping  steam. 

The  effect  produced  by  electrified  globules  of  water 
coalescing  is  to  raise  the  potential  of  the  cloud.  To  prove 
this,  let  us  consider  what  happens  when  n  drops,  of  radius  r 
and  at  potential  v,  coalesce  into  one  of  radius  R  and  at  po- 
tential F.  Since  the  volumes  and  charges  remain  the  same 
and  the  capacity  of  a  raindrop  is  approximately  equal  to 
its  radius,  we  have 

(4/3)7r.R3:=fl,(4/3)7rr3,  and  VE  =  nvr. 

Hence,  R3  =  nr3,  and 


LIGHTNING  CONDUCTORS 


247 


We  have  therefore 

V3  =  nW,  or  V  =  n2/3v. 

Hence  the  potential  of  the  large  drop  is  n213  times  that  of 
the  smaller  drops,  and  the  stress  F '/R  at  its  surface  equals 
nl/3(v/r). 

To  obtain  some  idea  of  the  potentials  called  into  play 
let  us  consider  the  disruptive  voltages  between  large  spherical 
electrodes.  In  the  following  table  (see  Chapter  VIII)  x  de- 
notes the  minimum  distance  in  metres  between  the  equal 
spherical  electrodes  whose  radius  is  stated,  and  F  is  the  dis- 
ruptive pressure  in  kilo  volts. 


Radius. 

1  cm. 

10  cm. 

100  cms. 

1,000  cms. 

X 

F 

V 

V 

F 

0-05 

61 

163 

187 

191 

0-1 

— 

280 

370 

381 

0-5 

— 

604 

1,625 

1,860 

1-0 

— 

— 

2,795 

3,690 

5-0 

— 

— 

6,030 

16,200 

10-0 

— 

— 

— 

28,000 

50-0 

-      — 

— 

— 

60,000 

For  instance,  when  the  potential  difference  between  two 
spherical  conductors  each  1  metre  in  radius  and  1  metre 
apart  attains  the  value  of  2,795  kilo  volts,  then,  under  normal 
atmospheric  conditions  the  air  between  them  will  be  broken 
down.  The  blank  spaces  in  the  above  table  refer  to  cases 
where  brush  discharges  ensue  before  the  disruptive  discharge. 
The  values  of  the  disruptive  voltages,  in  these  cases,  depend 
on  the  rate  at  which  the  ionization  of  the  air  surrounding  the 


248        ELECTRIC  CABLES  AND  NETWORKS 

electrodes  is  proceeding.     It  is  probable  therefore  that  their 
values  can  only  be  roughly  predetermined. 

In  the  table  of  the  sparking  voltage  between  needle  points 
published  by  the  American  Institution  of  Electrical  Engin- 
eers and  quoted  on  p.  179  it  will  be  seen  that  24-4  cms.  is 
given  as  the  sparking  distance  for  100  kilo  volts  alternating 
pressure  or  for  141*4  kilovolts  direct.  From  the  above  table 
we  see  that  163  kilovolts  will  spark  across  5  cms.  when  the 
radius  of  the  electrodes  is  10  cms.  and  consequently  the 
distance  between  their  centres  is  25  cms.  It  will  thus  be 
seen  that  if  we  suppose  the  spherical  electrodes  to  shrivel  up 
into  minute  spheres  having  the  same  centres  as  the  original 
electrodes  an  appreciably  smaller  voltage  will  suffice  to  break 
down  the  dielectric. 

Lightning    flashes  have  been   observed  more 

potential      than  two  miles  long  and  the  potential  differences 
difference 

required       required  previous  to  the  discharge  must  be  con- 
for  a 

lightning  siderable.  If  we  assumed  that  a  mean  electric 
stress  of  about  100  kilovolts  per  inch  is  necessary, 
then  about  13,000,000  kilovolts  would  be  required  to  pro- 
duce a  flash  two  miles  long.  This  number  fixes  a  superior 
limit  to  the  value  of  the  voltage  necessary  to  produce  this 
flash.  If  the  flash  were  to  occur  in  dry  clear  weather  between 
a  cloud  two  miles  high  and  the  earth,  and  the  air  between 
the  two  was  not  appreciably  ionized,  the  pressure  required 
might  possibly  be  about  10,000,000  kilovolts.  As  however, 
in  England  at  least,  lightning  flashes  practically  always 
occur  during  rain  or  hail  storms,  it  is  probable  that  a  much 
smaller  voltage  suffices.  We  have  already  seen  that  on  a 
clear  day  the  voltage  at  a  height  of  4,000  metres  is  usually 
at  least  1 50  kilovolts.  During  a  thunderstorm  it  is  probably 
at  times  much  higher.  As  a  first  rough  approximation  we 
may  conclude  that  the  voltage  between  an  ordinary  thunder- 


LIGHTNING  CONDUCTORS 


249 


The  A 
flash 


cloud  and  the  earth,  immediately  before  a  discharge,  lies 
in  value  between  100  and  1,000,000  kilo  volts. 

Sir  Oliver  Lodge  divides  lightning  flashes 
roughly  into  two  main  classes  which  he  calls 
the  A  and  the  B  flash  respectively.  These  flashes  produce 
very  different  effects  and  it  is  necessary  to  distinguish  care- 
fully between  them.  The  A  flash  is  illustrated  in  Fig.  64.  In 
this  case  the  differ- 
ence of  potential 
between  the  cloud 
and  the  earth  gradu- 
ally increases  until 
the  air  between  them 
breaks  down  owing 
to  the  great  electric 
stress  to  which  it  is 
subjected  and  a 
disruptive  discharge 
ensues  which  dimin- 
ishes appreciably  the  potential  difference  between  the  cloud 
and  the  earth.  The  distinguishing  characteristic  of  the  A  flash 
is  the  previous  gradual  building  up  of  the  voltage  between 
the  cloud  and  the  earth.  Immediately  before  the  flash 
occurs  the  potential  gradient  at  all  points  on  earthed  con- 
ductors is  very  steep.  Round  these  points  the  air  is  being 
ionized  at  a  rapid  rate  and  the  stream  of  ionized  air  from  them 
forms  a  path  of  small  resistance  for  the  disruptive  discharge. 
Lodge  has  devised  the  following  simple  and  instructive 
experiment  (Fig.  65)  to  illustrate  the  phenomena  connected 
with  the  A  flash.  C  and  E  are  metal  plates  insulated  from 
one  another.  They  are  connected  with  the  terminals  of  a 
Wimshurst  f  rictional  machine  W,  and  a  Leyden  jar  L  is  placed 
as  a  shunt  between  the  plates  so  as  to  increase  the  intensity 


FIG.  64.— The  A  Flash. 


250       ELECTRIC  CABLES  AND  NETWORKS 

of  the  discharge  when  it  occurs.  Model  lightning  con- 
ductors consisting  of  metallic  knobs  of  various  sizes  and 
shapes  on  conducting  supports  are  placed  on  the  lower  plate. 
We  may  consider  that  C  represents  the  cloud  and  E  the  earth. 


FIG.  P>5. — Model  illustrating  the  laws  governing  the  A  Flash. 

On  turning  the  handle  of  the  Wimshurst  machine  a  dif- 
ference of  potential  is  gradually  established  between  C  and 
E.  As  soon  as  the  potential  gradient  at  a  point  of  any  of 
the  conductors  exceeds  the  dielectric  strength  of  the  air 
between  the  plates  there  will  either  be  a  disruptive  spark 
between  the  conductor  and  <7,  or  there  will  be  a  brush  dis- 
charge from  the  conductor.  If  there  be  only  two  model  con- 
ductors on  E,  and  if  the  centre  of  the  small  knob  of  one  be 
closer  to  G  than  the  centre  of  the  large  knob  of  the  other,  it 
will  in  general  protect  it,  that  is,  the  disruptive  discharge 
will  take  place  between  the  smaller  knob  and  (7,  even  when 
the  minimum  distance  between  the  small  knob  and  G  is 
very  appreciably  greater  than  that  between  the  large  knob 
and  G. 

It  is  found  that  the  resistance  of  the  supporting  pieces 
connecting  the  knobs  with  the  lower  plate  has  very  little 


LIGHTNING  CONDUCTORS 

effect  on  their  liability  to  be  struck.  For  instance,  when  the 
stands  are  of  metal  and  a  damp  cloth  is  placed  between  the 
stand  of  the  small  knob  and  the  lower  plate,  the  small  knob 
is  still  struck  even  although  the  resistance  of  its  connexion 
with  the  lower  plate  is  hundreds  or  thousands  of  times 
greater  than  that  of  the  larger  knob. 

With  the  arrangement  shown  in  Fig.  65,  model  pointed 
conductors  are  so  effective  in  dissipating  the  charge  that  it  is 
almost  impossible  to  obtain  a  spark  at  all  when  they  are 
used.  If  a  lighted  gas  burner  be  placed  on  the  lower  tray 


FIG.  66.— Type  of  B  Flash.     The  A  Flash  between  the  clouds  causes 
the  B  Flash  to  the  earth. 

it  will  as  a  rule  protect  the  knobs,  the  spark  readily  passing 
to  the  flame  through  the  heated  products  of  combustion. 

If  the  top  tray  be  replaced  by  a  sieve  into  which  water 
is  poured,  it  is  impossible  to  obtain  a  spark  at  all. 

The  B  flash  which  is  caused  by  an  impulsive 
rush  of  electricity  occurs  when  the  difference  of 
potential  between  the  cloud  and  the  earth  is  established 
almost  instantaneously.  There  are  several  varieties  of  this 
flash.  In  Fig.  66,  for  example,  a  discharge  between  two 
clouds  alters  by  electrostatic  induction  the  potential  differ- 


The  B 

flash 


252        ELECTRIC  CABLES  AND  NETWORKS 

ence  between  another  cloud  and  the  earth,  and  this  voltage 
being  greater  than  the  air  can  withstand,  we  have  a  B 
flash  between  the  cloud  and  the  earth. 

An  experimental  illustration  of  this  flash  is  shown  in  Fig. 
67.  As  in  Fig.  65,  C  and  E  are  two  sheets  of  metal  represent- 


FIG.  67. — Model  illustrating  the  action  of  the  B  Flash. 

ing  a  cloud  and  the  earth,  L  and  M  are  two  Ley  den  jars 
which  we  suppose  to  be  placed  on  a  badly  insulating  wooden 
table.  Their  inner  coatings  are  connected  with  the  discharge 
knobs  D  of  a  Wimshurst  machine  W,  and  the  outer  coatings 
are  in  metallic  connexion  with  C  and  E. 

On  turning  the  handle  of  the  Wimshurst  machine,  the  inner 
coatings  are  brought  to  a  high  difference  of  potential,  and 
there  will  be  large  electrostatic  charges  induced  on  the  out- 
side coatings  of  the  jars  at  these  high  potentials.  When  a 
spark  occurs  at  D  the  potentials  of  the  inner  coatings  will 
be  equalized,  probably  by  an  oscillating  discharge,  and  the 
potential  difference  between  the  outside  coatings  of  L  and  M , 
owing  to  the  large  charges  on  them  of  equal  and  opposite 
sign,  will  attain  a  high  value.  Hence  also  the  potential 
difference  between  C  and  E  (Fig.  67)  which  are  in  metallic 


LIGHTNING  CONDUCTORS 


253 


connexion  without  the  outside  coatings  will  be  high,  and  if 
the  height  of  G  above  the  model  lightning  conductors  be 
not  too  great  there  will  be  a  spark  discharge.  Before  the 
spark  occurs  at  D,  the  potential  difference  between  the  plates 
is  very  small,  as  each  is  practically  at  earth  potential,  for  the 
inductive  effects  produced  by  the  equal  and  opposite  charges 
on  the  inner  coatings  of  the  Ley  den  jars  practically  neutral- 
izes the  effects  produced  by  the  outer  coatings.  But  when 
the  spark  occurs  at  D  the  potential  difference  between 
G  and  E  is  altered  practically  instantaneously,  the  spark 
between  them  is  therefore  of  the  B  type. 

In  this  case  the  ac- 
tion of  the  model  light- 
ning conductors  is  quite 
different  to  their  action 
in  the  preceding  case 
(Fig.  65).  In  Fig.  67, 
for  example,  where  the 
cone  and  the  tops  of  the 
knobs  are  all  of  the 
same  height  the  B  spark 
takes  place  between  C 
and  one  or  other  of 
these  conductors.  If 

one  be  placed  slightly  nearer  to  the  upper  plate  than  the 
others  it  will  protect  them. 

Other  varieties  of  the  B  flash  are  illustrated  in  figs.  68  and 
69.  An  experimental  illustration  of  these  cases  is  shown  in 
Fig.  70.  When  a  spark  takes  place  between  the  discharge 
knobs  D,  a  B  flash  will  pass  between  the  highest  lightning 
conductor  and  C  provided  that  the  distance  between  them 
be  not  too  great.  The  shape  of  the  end  of  the  lightning 
conductor  is  quite  immaterial  in  this  case,  the  protective 


FIG.  68. — A  second  type  of  B  Flash. 


254        ELECTRIC  CABLES   AND   NETWORKS 


action  being  quite  different  to  that  which  occurs  with  A 
flashes.  There  is  no  time  for  the  ionization  of  the  air  which 
takes  place  at  points  to  prepare  a  path  of  small  resistance 
for  the  discharge.  The  rush  of  electricity  apparently  always 
takes  place  across  the  shortest  path.  Lodge  compares 
the  paths  in  the  steady  stress  and  in  the  impulsive  rush  cases 
to  the  paths  taken  down  a  hill  side  by  a  gentle  stream  of 
water  and  by  an  avalanche  respectively. 


FIG.  69. — A  third  type  of  B  Flash.     The  A  Flash  from  a  cloud  to  the 
chimney  stack  causes  the  B  Flash  from  a  neighbouring  cloud. 

As  in  the  case  of  the  A  flash  it  is  found  that  the  absolute 
values  of  the  resistances  of  the  lightning  conductors  them- 
selves have  little  effect  on  their  protective  qualities. 

If  we  replace  the  top  sheet  of  metal  in  Figs.  67  and  70  by 
a  sieve  into  which  water  is  poured,  sparks  still  ensue.  The 
flashes,  like  those  which  occur  during  thunderstorms,  are 
sometimes  very  long  and  very  irregular.  They  seem  to 
make  use  of  the  rain  drops  as  stepping  stones. 

As  thunderstorms  in  this  country  are  nearly  always  ac- 
companied by  rain,  it  is  highly  probable  that  most  of  the 
flashes  which  occur  belong  to  the  impulsive  rush  case.  In 
the  majority  of  cases  also  it  is  probable  that  the  discharge 
is  oscillatory,  for  we  know  both  by  theory  and  experiment 


LIGHTNING  CONDUCTORS 


255 


that  the  spark  discharge  of  a  condenser  is  oscillatory  pro- 
vided that  the  resistance  of  the  path  of  the  discharge  be  not 
above  a  certain  value. 

The  main  function  of  a  lightning  rod  is  to 
dissipate  the  energy  stored  in  the  lightning  flash 
harmlessly,  and  so  prevent  it  from  doing  damage  to  neigh- 
bouring objects.  Hence  the  conductor  must  not  be  too 
small  in  diameter  or  it  will  be  deflagrated  by  the  discharge. 


Lightning 
rods 


FiG.  70. — Model  illustrating  second  type  of  B  Flash. 

A  subsidiary  function  is  to  equalize  the  potential  between 
the  thunder  cloud  and  earth  by  the  "  silent  discharge  " 
taking  place  from  all  points  on  the  conductor.  This  action 
is  probably  less  energetic  than  the  discharging  action  of 
certain  kinds  of  trees — for  instance,  fir  trees.  Statistics 
prove  that  the  cutting  down  of  extensive  fir  forests  in  certain 
parts  of  Europe  has  led  to  a  considerable  increase  in  the 
number  of  destructive  lightning  flashes  experienced  in  those 
districts.  It  is  probable,  therefore,  that  in  towns  where 
numerous  lightning  conductors  with  multiple  points  on 
them  are  employed,  they  will  have  the  effect  of  diminishing 
the  average  number  of  the  lightning  flashes  that  occur, 


256        ELECTRIC  CABLES   AND  NETWORKS 

The  When  the  discharge  is  oscillatory  it  must  have 

Cinrthe  an  exceedingly  high  frequency,  and  when  it  is 
non-oscillatory  the  discharge  is  over  in  a  very 
small  fraction  of  a  second.  In  either  case  we  know  from 
theory  that  the  current  in  the  lightning  conductor  flows  so 
that  the  magnetization  in  the  metal  of  the  conductor  is 
a  minimum  (c/.  Chapter  II,  p.  43).  It  is  therefore  prac- 
tically confined  to  a  thin  layer  of  the  metal  on  the  outer 
surface  of  the  conductor.  In  calculating  the  resistance  of 
the  path,  therefore,  we  must  not,  as  in  electric  light  wiring, 
proceed  on  the  assumption  that  the  current  density  is  uniform 
over  the  cross  section. 

Lord  Rayleigh  has  shown  that  when  the  frequency  is  very 
high  the  resistance  R  and  the  self  -inductance  L  of  a  cylindri- 
cal rod  for  a  symmetrical  flow  of  current,  obeying  the  sine 
law,  are  given  by 

and   L 


In  these  equations  I  denotes  the  length  of  the  conductor,  a 
its  radius,  p  the  resistivity  in  absolute  units,  ^  the  permea- 
bility of  the  metal,  /  the  frequency,  and  A  a  constant  depend- 
ing on  the  dimensions,  etc.,  of  the  return  circuit.  All  the 
quantities  in  the  equations  are  in  C.G.S.  units. 
Numerical  ^et  us  suPPose  that  the  lightning  conductor 

example  jg  a  Cylin^rical  copper  rod  100  metres  long  and 
1  centimetre  in  diameter.  In  this  case  1=  10,000,  a  =  0-5, 
p  =  1,600  approximately,  and  /j,=  1.  We  shall  suppose  also 
that  the  frequency  is  1,000,000,  so  that  /=106.  Sub- 
stituting these  values  in  the  formula  for  R  we  find  that 
12  =  (104/0-5)y  1,600  x!06=8  xlO8  absolute  units  =  0-8  of 
an  ohm.  The  resistance  to  a  flow  of  direct  current  would  be 
0-02  of  an  ohm.  Hence  the  resistance  of  the  lightning 
conductor  to  an  impulsive  rush  of  electricity  is  forty  times 
as  great  as  that  which  it  would  offer  to  direct  currents  or 


LIGHTNING  CONDUCTORS  257 

alternating  currents  of  the  frequencies  used  for  electric 
lighting. 

If  the  conductor  had  been  made  of  iron,  all  the  dimensions 
remaining  the  same,  and  if  we  take  the  resistivity  of  iron  as 
nine  times  that  of  copper  and  assume  that  its  average  per- 
meability under  the  given  conditions  is  100,  then,  R  will 
equal  24  ohms,  and  the  resistance  with  direct  current  will  be 
0-18  of  an  ohm  which  is  less  than  the  hundredth  part  of  the 
apparent  resistance  to  the  alternating  impulsive  rushes. 

It  follows  from  Rayleigh's  formula  that  the  inductance 
of  the  conductor  is  lA-\-R/2jrf  .  Hence  the  reactance  is 
27rflA-{-R.  In  most  cases  R  will  be  very  small  compared 
with  2-7T/L4,  and  hence  the  inductance  and  reactance  of 
lightning  conductors  is  practically  independent  of  the 
material  of  which  they  are  made. 

Rayleigh's  formulae  show  that  the  greater  the  radius  of  a 
cylindrical  rod  the  smaller  will  be  its  resistance  and  induct- 
ance. As  the  radius  of  the  rod  increases,  however,  the 
greater  will  be  the  ratio  of  the  apparent  resistance  Ra  with 
alternating  currents  to  the  resistance  Rd  with  direct  currents, 


for  Ra  /  Rd  =  {(l  I  a)Vf}/(Pl/7ra*)=7ratfp.  Hence 
the  ratio  Ra/Rd  varies  directly  as  the  radius  of  the  rod,  but 
the  absolute  value  of  Ra  diminishes  as  a  increases. 

It  is  to  be  remembered  that  the  longer  the  conductor,  or 
the  greater  its  resistance,  the  lower  will  be  the  frequency  of 
the  oscillations  set  up  by  the  lightning  flash.  We  have  also 
to  remember  that  in  calculating  the  values  of  R  no  account 
has  been  taken  of  the  energy  lost  by  radiation  into  space, 
which  at  these  high  frequencies  is  probably  appreciable. 

If  a  piece  of  wire  is  placed  sufficiently  close 
Side  flash 

to  a  lightning  rod,  part  of  the  charge  will  leave 

the  rod  and  travel  along  the  piece  of  wire  as  the  reactance 
of  the  divided  path  is  less  than  the  path  in  the  conductor 

s 


258        ELECTRIC  CABLES   AND  NETWORKS 

alone.  This  explains  the  phenomenon  of  side  flash  which 
is  often  observed  when  an  object  is  struck  by  lightning. 
The  potential  differences  existing  between  various  parts  of 
a  lightning  conductor  when  it  is  struck  are  obviously  very 
high  and  hence  the  electrostatic  field  round  it  is  very  intense. 
The  electric  stresses  ionize  the  air  round  the  conductor  and 
so  a  spark  readily  ensues  to  any  neighbouring  conductor. 
In  setting  up  lightning  conductors  this  tendency  to  side 
flash  has  to  be  remembered,  as  sparks  due  to  this  cause  can 
ignite  escaping  gas  and  thus  set  fire  to  buildings.  It  has 
often  been  noticed  that  when  a  lightning  rod  is  struck  a 
peculiar  noise  is  heard  not  unlike  the  pouring  of  water  on 
a  fire,  and  electric  sparks  are  emitted  from  bodies  in  the 
neighbourhood.  These  phenomena  are  probably  caused 
by  brush  discharges  due  to  the  breaking  down  of  the  air  by 
the  electrostatic  stresses  set  up  during  the  discharge. 

The  Lightning  Research  Committee,  appointed 

Research      by  the  Royal  Institution  of  British  Architects 
and  the  Surveyors'  Institution  in  1901,  have  in 
their  report  made  the  following  practical  suggestions. 

1.  Two  main  lightning  rods,  one  on  each  side  should  be 
provided,  extending  from  the  top  of  each  tower,  spire  or 
high  chimney-stack  by  the  most  direct  course  to  earth. 
The  diagrams  shown  in  Fig.  71  illustrate  this  sugges- 
tion. 

In  Y,  which  is  the  usual  method,  the  conductor  follows 
the  outline  of  the  building.  In  this  case  there  is  a  tendency 
for  the  discharge  to  leave  the  conductors  at  the  bends,  as 
it  always  tends  to  make  a  path  or  paths  for  itself  in  addition 
to  that  provided  by  the  lightning  rod,  so  that  the  reactance 
of  all  of  them  in  parallel  may  be  a  minimum.  It  thus  some- 
times breaks  away  the  brickwork,  and  in  some  cases  the 
mechanical  forces  called  into  play  break  the  conductor 


LIGHTNING  CONDUCTORS  259 

itself.  The  Research  Committee  recommend  the  method 
illustrated  in  X  (Fig.  71),  where  the  conductor  is  kept  away 
from  the  building  by  suitable  holdfasts,  which  may  be  made 
of  iron. 

It  seems  to  the  author  that  the  method  X  recommended 
is  excellent  for  getting  round  sharp  corners,  or  in  cases  where 
there  is  danger  from  side  flash  owing  to  the  presence  of 
neighbouring  conductors.  In  general,  however,  when  there 


FIG.  71. — X  is  the  method  of  fixing  lightning  conductors  recommended 
by  the  Lightning  Research  Committee. 


is  a  straight  run  for  the  conductor  there  is  no  need  to  keep 
it  away  from  the  surface  of  the  wall. 

2.  Horizontal  conductors  should  connect  all  the  vertical 
rods  (a)  along  the  ridge,  (b)  at  or  near  the  ground  line. 

This  recommendation  of  the  Committee  was  probably 
suggested  to  obviate  the  risks  of  side  flash  from  one  con- 
ductor to  the  other  and  as  a  partial  protection  also  for  the 
space  between  the  two. 


260        ELECTRIC   CABLES  AND  NETWORKS 

3.  The  upper  horizontal  conductor  should  be  fitted  with 
aigrettes  or  points  at  intervals  of  20  or  30  feet. 

4.  Short  vertical  rods  also  should  be  erected  along  minor 
pinnacles,  and  connected  with  the  upper  horizontal  con- 
ductor. 

5.  All  roof  metals  such  as  finials,  ridging,  rain  water  and 
ventilating  pipes,  metal  cowls,  lead  flushing,  gutters,  etc., 
should  be  connected   with  the  horizontal  conductors. 

6.  All  large  masses  of  metal  in  the  building  should  be  con- 
nected with  the  earth,  either  directly  or  by  means  of  the 
lower  horizontal  conductor. 

7.  Where  roofs  are  partially  or  wholly  metal-lined  they 
should  be  connected  with  the  earth  by  means  of  vertical 
rods  at  several  points. 

8.  Gas  pipes  should  be  kept  as  far  away  as  possible  from 
the  positions  occupied  by  lightning  conductors,  and  as  an 
additional  protection  the  service  mains  of  the  gas  meter 
should  be  metallically  connected  with  house  services  leading 
from  the  meter. 

Many  useful  suggestions  will  also  be  found  in  the  Report 
issued  by  the  Lightning  Rod  Conference  held  in  1882. 
Some  of  the  rules  given  in  this  report,  however,  have  to  be 
amended  as  they  proceed  on  the  erroneous  assumption  that 
a  lightning  flash  will  follow  the  path  of  minimum  resistance 
in  exactly  the  same  way  that  a  steady  direct  current  would. 
The  end  of  the  lightning  conductor  is  usually 
connected  with  a  copper  plate  embedded  in 
moist  earth  in  the  neighbourhood  of  the  building.  If  none 
of  the  earth  in  the  immediate  neighbourhood  of  the  con- 
ductor is  moist,  it  is  advisable  to  dig  a  pit  about  6  feet  deep 
in  which  the  sheet  of  copper  about  a  square  yard  in  area  and 
one-eighth  of  an  inch  thick  should  be  placed  and  then 
surrounded  with  charcoal  or  pulverized  carbon.  The  ends 


LIGHTNING  CONDUCTORS  261 

of  the  carbons  used  in  arc  lamps  do  excellently  for  this 
purpose.  Coke  is  sometimes  employed,  but  its  use  is 
objectionable  owing  to  the  chemical  and  electrolytic  effects 
produced  in  the  copper.  The  pit  should  not  be  quite  filled 
up  with  earth,  so  that  there  may  be  a  sufficient  depression 
on  the  surface  over  the  pit  to  catch  the  rain  during  a 
thunderstorm  and  thus  keep  the  earth  in  the  neighbour- 
hood of  the  plate  moist. 

The  resistance  of  the  "  earth  "  is  measured  by  finding,  by 
a  Wheatstone's  bridge  or  otherwise,  the  resistance  between 
the  conductor  and  any  neighbouring  water  pipe.  On  a  dry 
day  if  this  resistance  be  not  greater  than  100  ohms  the 
"  earth  "  may  be  considered  satisfactory.  Accurate  mea- 
surements of  this  resistance  are  neither  possible  nor 
necessary.  In  the  neighbourhood  of  towns  supplied  with 
electric  light  or  tramways  a  permanent  deflection  is  often 
obtained  on  the  galvanometer  owing  to  a  leakage  current 
from  some  of  the  supply  networks.  Instead  of  using  a 
special  "  earth  "  it  is  sometimes  convenient  to  connect  the 
end  of  the  lightning  conductor  with  the  water  mains. 

Tubular  Mr.  Killingworth  Hedges'  "  tubular  earth  " 

earth  can  often  be  advantageously  used.  It  consists  of 
a  hollow  perforated  steel  spike  filled  with  granulated  carbon 
and  driven  into  moist  earth.  The  lightning  conductor  is 
taken  to  the  bottom  of  the  tube.  The  earth  in  the  neigh- 
bourhood of  this  device  can  easily  be  kept  moist  by  con- 
necting it  with  the  nearest  rain-water  pipe.  In  this  case 
the  earth  resistance  is  negligibly  small. 

The     lightning     conductors     used     in     this 
The  metal 

of  the        country  are  generally  made  of  copper.     Either 
conductor 

copper  tape  or  copper  wire  rope  is  employed. 

In  the  former  case  the  section  is  usually  }  inch  broad  by 
•J-  inch  thick,  this  size  being  found  ample  in  practice.  In 


262        ELECTRIC  CABLES  AND  NETWORKS 

the  latter  case  the  rope  is  usually  J  inch  in  diameter.  If 
smaller  sized  conductors  are  used  there  is  a  risk  of  them 
being  deflagrated  by  a  severe  lightning  flash. 

In  those  climates  where  there  is  little  risk  of  the  con- 
ductor being  corroded  either  by  the  moisture  or  by  chemical 
fumes,  galvanized  soft  stranded  iron  rope  is  the  most  suitable 
lightning  conductor.  The  higher  specific  heat  of  an  iron 
conductor  compensates  for  its  smaller  density,  and  so  its 
higher  melting  point  enables  it  to  get  rid  of  a  larger  amount 
of  the  electrical  energy  of  the  flash  than  a  copper  conductor 
of  the  same  dimensions. 

The  elevation  rod  or  top  of  a  lightning  con- 
elevation      ductor  ought  to  be  the  highest  point  of  the 
building,  but  there  is  no  necessity  to  have  it 
more  than  about  a  foot  taller  than  the  summit  of  a  pinnacle 
or  the  brickwork  of  a  chimney.     Four  or  five  well  gilded  or 
platinized  "  points  "  should  be  attached  to  the  elevation 
rod. 

Town  There  is  not  much  danger  of  town  houses 

houses  being  struck  by  lightning,  as  the  numerous 
gutters,  ventilating  and  rain-water  pipes  afford  them  con- 
siderable protection.  Occasionally  metallic  bonds  are  used 
to  connect  the  various  sections  of  rain- water  pipes,  and  thus 
ensure  their  metallic  continuity  and  so  guard  against 
damage  by  lightning.  As  most  fire  insurance  policies 
issued  in  England  cover  damage  done  by  lightning,  these 
precautions  are  seldom  taken  for  ordinary  town  houses. 
Important  buildings  are  usually  elaborately  protected  by 
lightning  conductors.  Even  with  very  elaborate  systems, 
however,  possible  dangers  arise  from  side  flash  from  the 
conductors  to  neighbouring  gas  pipes  or  stove  pipes.  The 
Hotel  de  Ville  at  Brussels,  which  is  protected  by  a  very 
complete  network  of  wires,  had  a  narrow  escape  from 


LIGHTNING  CONDUCTORS  263 

being  burned  down  during  a  thunderstorm,  as  a  spark  from 
one  of  the  lightning  conductors  to  a  neighbouring  piece  of 
metal  set  fire  to  gas  which  had  escaped  from  a  leak  in  a 
gas  pipe. 

Lightning  Heavy  damp  soils  such  as  loam  are  particu- 
fataiities  jarjy  u'able  to  be  struck  by  lightning  flashes. 
The  most  frequent  fatalities  in  this  country  from  lightning 
happen  to  people  standing  under  trees  which  are  struck, 
the  lightning  "  side  flashing  "  from  the  tree  to  the  person 
whose  body,  or  clothes  if  wet,  forms  a  good  conductor. 
Trees  whose  roots  are  near  the  water  are  particularly  liable 
to  be  struck.  Again  a  person  in  the  centre  of  a  field  or 
crossing  the  brow  of  a  hill  might  possibly  be  struck,  as  he 
would  be  the  highest  object  in  the  neighbourhood.  Horses, 
cattle,  and  sheep,  especially  when  steam  is  rising  from  them 
owing  to  their  being  overheated,  are  sometimes  struck.  Deer 
in  public  parks  are  frequently  killed  owing  to  their  habit 
of  congregating  under  trees  during  thunderstorms.  In 
America  wire  ropes  are  of  ten  used  to  hang  clothes  on  to  dry 
after  being  washed.  Several  fatalities  occur  every  year  to 
people  taking  the  clothes  off  these  ropes  at  the  beginning 
of  a  thunderstorm. 

REFERENCES. 

R.  Anderson,  Lightning  Conductors. 

Sir    Oliver    Lodge,    "  On    Lightning,    Lightning    Conductors     and 

Lightning  Protectors."     Journ.  of  the  Inst.  of  EL  Engin.,  vol. 

viii,  p.  386,  1889. 

Sir  Oliver  Lodge,  Lightning  Conductors  and  Lightning  Guards. 
Killingworth  Hedges,   Modern  Lightning  Conductors. 
Melsens,  Des  Paratonnerres  d   pointes,  d  conducteurs  et  d  raccorde- 

ments  terrestres  multiples  ;   description  detaillee  des  paratonnerres 

etablis  sur  r hotel  de  ville  de  Bruxelles.     1877. 
F.    Linke,    "  Luftelektrische     Messungen    bei    12    Ballonfahrten." 

(Abh.  der  Koniglichen  Gesell.   der  Wiss.  zu  Gottingen,  Math- 

Phys.  Klasse,  Neue  Folge,  Band  iii,  1904.) 


264        ELECTEIC  GABLES   AND  NETWORKS 

C.  Chree,   "  A  Discussion  of  Atmospheric  Electric  Potential  Results 

at  Kew,  from  selected  days  during  the  seven  years  1898-1904." 

Phil.  Trans.,  Series  A.,  vol.  206,  p.  299,  1906. 
Lord  Rayleigh,   "  On  the  Self -Induction  and  Resistance  of  Straight 

Conductors."     Phil.  Mag.  [5]  vol.  21,  p.  381,  1886,  or  Scientific 

Papers,  vol.  ii.,  p.  493. 
Report  of  the  Lightning  Research  Committee.     Journal  of  the  Royal 

Institute  of  British  Architects    [111]  vol.  12,  p.  405,  1904. 


INDEX 


"  A  "  flash,  249 

Air,  dielectric  strength  of,   177, 
182 

sparking  distances  in,  179 
Algermissen,  J.,  177 
Allgemeine  Elektricitats  Gesell- 

schaft,  228 
Alternating  currents, 

high  frequency,  43,  256 
American  rules,  178 
American  Standardization  Com- 
mittee, 31 
Anderson,  R.,  263 
Annealed  copper,  21 
Appleyard,  R.,  60,  64 
Argon,  dielectric  strength  of,  182 
Arresters,  continuous,  232 

electrolytic,  235 

horn,  227 

intermittent,  226 

lightning,  225  et  seq. 

multiple  gap,  230 

water  jet,  233,  235 
Atmospheric   electricity,    243    et 

seq. 

Atmospheric  potential  gradient, 
224 

"  B  "  flash,  249,  251 

B.  and  S.  Gauge,  13 

B.W.G.,  13 

Baur,  C.,  64 

Benton,  J.  R,  22 

Birmingham  Wire  Gauge,  13,  15 

Black,  G.  L.,  159 

Blavier's  test,  158 

Board  of  Trade  Regulations,  68, 

97,  125 
Booster,  90 
Bose,  M.  von,  45 
Breaks,  locating,  145 


Bridge  method  of  testing,  155 
British  Legal  Standard,  13 
British  Standard  Radial  Thick- 
nesses, 203 

Brown  and  Sharpe  Gauge,  13,  15 
Bulk  modulus,  20 

Cables,  grading,  187  et  seq. 

insulation  resistance  of,   58 

mass  of,  15,  37,  40 

resistance  of,  39,  42 

stranded,  33 
Cadmium  cell,  55 
Calorie,  5 

Campbell,  A.,  52,  196 
Centre  of  gravity  of  load,  85 
Centres,  distributing,  70 
Charing  Cross  Company,  228 
Chree,  C.,  245,  246,  264 
Circular  mil,  12 
Clark,  W.  S.,  64 
Collie,  J.  N.,  182 
Composite  dielectrics,  173,  207 
Concentric  main,  43 

electric  stresses  in,  188 

grading  of,  191  et  seq. 

suitable  dimensions  for,  190 

temperature  gradient  in,  212. 
Condenser,  spherical,  167 
Conductance,  12 
Conductivity,  19  et  seq. 
Conductors,  in  parallel,  7 

in  series,  6 

lightning,  243  et  seq. 
Copper,  annealed,  21 

density,  23 

elastic  constants,  21 

hard  drawn,  21 

temperature  coefficient  of,  27 

useful  data  for,  45 
Current  density,  40 


265 


266 


INDEX 


Current,  excessive,  73 
high  frequency,  43,  256 
in  lightning  conductors,  256 
permissible,  40 

Data  for  calculations,  45 
Density  of  copper,  23 

standard,  24 

Dewar,  J.,  11,  16,  27,  32,  46 
Dielectrics,  composite,  173 
Dielectric  strength,  163  et  seq. 

coefficients  for,  176 

of  air,  177 

of  eolotropic  solids,  183 

of  gases,  180,  182 

of  iso tropic  solids,  183 
Disruptive  discharge,  166 
Distributing  centres,  70 
Duddell  currents,  225 
Dusaugey,  239 

Earth  currents,  energy  expended 
in,  123 

Earth  faults,  112,  116,  140 

Earth,  in  house  wiring,  140 
in  middle  main,  147 
in  negative  outer,  149 

Earth  lamps  detector,  107 

Earthing    lightning    conductors, 
260 

Elastic  constants,  19 

Electric  intensity,  166 
stress,  164 

Electricity,  atmospheric,  243 

Electrolytic    arresters,  235,    238 

Electrostatic  voltmeter  method, 
106 

Elevation  rod,  262 

Endosmosis,  121 

Energy  expended  in  earth  cur- 
rents, 123 

Engineering     Standards     Com- 
mittee, 103,  203 

Eolotropic  solids,  183 

Evershed  and  Vignoles,  104 


Fall  of  potential  method,  153 
Faraday,  M.,  164 
Fault  resistance,  112 

measurement  of,  100,  113,  133 
Faults  in  networks,  159 

locating,  145  et  seq. 

locating,  by  flashing,  147 
final  methods,  153 
general  methods,  151 
Feeding  centres,  73,  78  et  seq. 

for  straight  main,  87 
Fernie,  F.,  151,  159 
Fisher,  H.  W.,  184 
Fitzpatrick,  T.  C.,  23,  46 
Flashing,  147 

Fleming,  J.  A.,  11,  16,  27,  32,  46, 
159 

Garrard,  C.  C.,  227,  229 

Gases,  dielectric  strength  of,  180 

Gauges,  13 
tables  of,  15 

Gauss's  Hypergeometric  Series, 
205 

Glazebrook,  R.  T.,  26 

Gotti,  O.  Li,  92 

Grade  of  insulation,  56 

Gradient,  potential,  165 

Grading  of  cables,  187  et  seq. 

Graphical    construction    for   po- 
tentials, 117 

Gray,  T.,  184 

Groves,  W.  E.,  135 

Guard  wire,  Price's,  56 

Gutta,  64 

Hard  drawn  copper,  21 

Heat,  effects  on  dielectric  of,  214 

Heating  of  bare  conductors,  220 

of  cables,  211  et  seq. 

of  wire,  5 

Hedges,  Killingworth,  261,  263 
Helium,  dielectric  strength,   182 
Herzog,  J.,  92 

High  frequency  alternating  cur- 
ents,  43,  256 


INDEX 


267 


High  pressure,  economy  of,  71 
Hobart,  H.  M.,  184 
Hobbs,  G.  M.,  179 
Hooke's  law,  19 
Hopkinson,  J.,  153 
Horn  lightning  arrester,  227 
House  wiring,  faults  in,  140 
Hydrogen,  dielectric  strength  of, 
182 

Induction  method,  156 

Institution  Rules,  57,  95 

Insulation  resistance,  49 
formula  for,  51 
measurement  of,  53 
of  house  wiring,  95  et  seq. 
of  three  wire  network,  119 
of  two  wire  network,  111 
Institution  rules  for,  57 
tables  of,  58,  61 

Insulation,  specific,  50 

Insulativity,  49  et  seq. 

Intensity,  electric,  166 

Iso tropic  bodies,  3 

dielectric  strength  of,  183 

Jackson,  R.  P.,  184,  239 
Jaeger  and  Kahle's  formula,  55 
Jona,  E.,  184,  189,  197,  205,  207, 

214 

Jona's  graded  cables,  197 
Joule's  law,  5 

Kelvin,  Lord,  246 

skin  effect,  44 

Thomson  and  Tait,  86 
Kelvin's  law,  67 
Kennelly,  A.  E.,  222 
Kew  observations,  245 
Kirchhoff's  first  law,  6 

second  law,  6 

Langan,  J.,  64 

Lay,  definition  of,  37 

effect  on  mass  of  cable  of,  37 
effect  on  resistance  of  cable  of, 
39 


Leak  in  middle  main,  122 

in  positive  outer,  122 
Leakage  currents,  125 

effect  on  grading  of,  199 
Leboucq,  M.,  92 
Lees,  C.  H.,  213 
Levi-Civita,  205 
Lightning    arresters,    228,    231, 

233 

Lightning    conductors,    243     et 
seq. 

currents  in,  256 

metal  of,  261 

method  of  fixing,  259 
Lightning  fatalities,  263 
Lightning  Research  Committee, 

258,  264 

Lightning  Rod  Conference,  260 
Lines  of  flow,  30 
Linke,  F.,  244,  263 
Load,  uniformly  distributed,    71 

centre  of  gravity  of,  85 
Lodge,  Sir  Oliver,  243,  249,  254, 
263 

Main,  concentric,  43 

feeding  centre  for  straight,  87 

grading,  188  et  seq. 

single  core,  169 

thermal  conductance  of,  217 
Mass  of  conductor, 

effect  of  "  lay  "  on,  37 

tables,  15 

Mass  resistivity,  24 
Matthiessen,  A.,  45 
Matthiessen's  Standards,  25 
Megger,  the  Evershed,  105 
Melsens,  263 

Microscopic  spark  lengths,  179 
Mie,  G.,  222 
Mil,  12 

circular,  12 
Minimum  heating  of  networks, 

7,  9,  124 

Minimum   insulation   resistance, 
58 


268 


INDEX 


Minimum  radial  thicknesses,  58, 

203 
Modulus,  bulk,  20 

Young's  20 
Monatomic  gases,  206 
Moore,  A.,  227 
Multiple  gap  arrester,  230 

Neon,  dielectric  strength  of,  206 
Networks,  distributing,  68  et  seq. 
Nodon  valve,  236 
Norden,  K.,  236 
Numerical  data,  45 

Oerlikon  Company,  227 
O'Gorman,  M.,  199,  207 
Ohmmeter,  103 
Ohm's  law,  4 

Oxygen,   dielectric   strength   of, 
182 

Patchell,  W.  H.,  228 
Peck,  J.  S.,  239 
Permissible  current,  40 
Perrine,  F.  A.  C.,  64 
Picou,  R.  V.,  222 
Pointed  conductor,  172 
Pollak  rectifier,  236 
Potential  gradient,  165 
Power  station, 

site  of,  85 

Preece,  Sir  William,  50,  64 
Pressure     at     consumer's     ter- 
minals, 68 
Price,  W.  A.,  56 

Ramsay,  Sir  W.,  182 
Raphael,  F.  C.,  16,  159 
Rayleigh,  Lord,  256,   264 
Rayner,  E.  H.,  64 
Rectifier,  Pollak,  236 

Nodon  valve,  236 
Regulation  of  potentials,  120 
Resistance  of  stranded  cables,  39 
Resistances,  in  parallel,  8 

in  series,  6 


Resistivity,  mass,  24 

volume,  9 

Resultant  electric  force,  166 
Rhodin,  J.,  27,  46 
Ring  mains,  81,  83,  84 
Rubber,  63 
Russell,  S.  A.,  64 
Russell,  A.,  108,   130,  135,   184, 

201,  207,  219,  222 
Ryan,  H.  J.,  184 

Safety  valves,  225  et  seq. 

continuous,  232 

intermittent,  226 
Schuster,  A.,  184 
Schwartz,  A.,  64,  159 
Searle,  G.  F.  C.,  21,  46,  213,  222 
Sections,  economical,  of   mains, 

74  et  seq. 

Seibt,  G.,  229,  239 
Shape  of  electrodes,  effect  of,  171 
Shear,  20 

Short  circuits,  locating,  143 
Side  flash,  257,  263 
Siemens  Bros.,  52 
Siemens  and  Halske,  227 
Single  core  main, 

grading  of,  191 

heating  of,  217 
Skin  effect,  44 
Skinner,  C.  E.,  184 
Societe   d'    Applications   Indus- 

trielles,  234 
Societe  de  Grenoble  et  Voiron, 

231 

Societe  de  Vizille,  233 
Specific  insulation,  50 
Spherical  condenser,  167 
Spherical  electrodes,  172 
Standard  wire  gauge,  13,  15 
Star  connexion,  7 

potential,  7 
Stark,  L.,  92 
Steimnetz,  C.  P.,  184 
Straight  main,  feeding  centre  for, 
87 


INDEX 


269 


Stranded  cables,  33  et  seq. 
Stranding,     effect     on     electric 

stresses  of,  205 
Swan,  J.  W.,  27,  46 

Tables,  gauges,  15 

mass  of  wires,  15 

maximum  stress,  176 

minimum  radial  thickness,  58 

minimum  insulation  resistance, 
58 

RD.  drop,  40 

permissible  currents,  40 

resistance  of  cables,  42 

section  of  cables,  40 

sparking  distances,  177,  179 
Taylor,  A.  M.,  135 
Temperature  coefficient, 

of  copper,  27 

of  gutta,  52 

of  metals,  33 

of  rubber,  52 
Temperature  gradient, 

effect  on  dielectric  stress,  214 

in  conductors,  216 


Temperature  measurement,  31 
Thermal  conductance,  217 

in  polycore  cables,  218 
Thomas,  P.  EL,  239 
Thomson     (Lord     Kelvin)     and 

Tait's  Nat.  Phil.  86 
Thomson,  J.  J.,  16 
Three  core  cable,  219 
Three  wire  potentials,  116 

graphical  construction  for,  117 
Tubular  earth,  261 
Turner,  H.  W.,  184 
Two  ammeter  method,  155 

Volume  resistivity,  9 
of  metals,  11 
section  variable,  10 

Wiring  rules,  95 

Young's  modulus,  20 
for  metals,  21 

Zenneck,  177 


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Elements  of  Electric  Lighting,  including  Electric  Generation,  Measurement, 
Storage,  and  Distribution.  Tenth  Edition,  fully  revised  and  new  matter 
added.  Illustrated.  12mo.,  cloth,  280  pp $!.£() 

Power  Transmitted  by  Electricity  and  Applied  by  the  Electric  Motor,  including 
Electric  Railway  Construction.  Illustrated.  Fourth  Edition,  fully  revised 
and  new  matter  added.  12mo.,  cloth,  241  pp .  .$2 .00 

AYRTON,  HERTHA.     The  Electric  Arc.     Illus.     8vo.,  cloth,  479  pp.  ...  Net,  $5 . 00 

—  W.  E.  Practical  Electricity.  A  Laboratory  and  Lecture  Course.  Illus- 
trated. 12mo.,  cloth,  643  pp $2 .00 

BIGGS,  C.  H.  W.  First  Principles  of  Electricity  and  Magnetism.  Illustrated. 
12mo.,  cloth.  495  pp $2.00 

BONNEY,  G.  E.  The  Electro-Plater's  Hand  Book.  A  Manual  for  Amateurs 
and  Young  Students  on  Electro-Metallurgy.  Fourth  Edition,  enlarged. 
61  Illustrations.  12mo.,  cloth,  208  pp $1 .20 

BOTTONE,  S.  R.  Magnetos  For  Automobilists,  How  Made  and  How  Used.  A 
handbook  of  practical  instruction  on  the  manufacture  and  adaptation  of 
the  magneto  to  the  needs  of  the  motorist.  Illustrated.  12mo.,  cloth, 
88  pp Net,  $1 .00 

Electric  Bells  and  All  about  Them.     12mo.,  cloth 50  cents 

Electrical  Instrument-Making  for  Amateurs.  A  Practical  Handbook.  En- 
larged by  a  chapter  on  "The  Telephone."  Sixth  Edition.  With  48  Illus- 
trations. 12mo.,  cloth 50  cents 

Electric  Motors,  How  Made  and  How  Used.  Illustrated.  12mo.,  cloth, 
168  pp 75  cents 

BOWKER,  WM.  R.  Dynamo,  Motor,  and  Switchboard  Circuits  for  Electrical 
Engineers:  a  practical  book  dealing  with  the  subject  of  Direct,  Alternating, 
and  Polyphase  Currents.  With  over  100  Diagrams  and  Engravings.  8vo., 
cloth,  120  pp Net,  $2.25 

BUSIER,  E.  T.  Questions  and  Answers  about  Electricity.  A  First  Book  for 
Beginners.  12mo.,  cloth 50  cents 

CARTER,  E.  T.  Motive  Power  and  Gearing  for  Electrical  Machinery;  a  treat- 
ise on  the  theory  and  practice  of  the  mechanical  equipment  of  power 
stations  for  electric  supply  and  for  electric  traction.  Second  Edition,  revised. 
Illustrated.  8vo.,  cloth,  700  pp Net,  $5 . 00 

CHILD,  CHAS.  T.  The  How  and  Why  of  Electricity:  a  book  of  information  for 
non-technical  readers,  treating  of  the  properties  of  Electricity,  and  how 
it  is  generated,  handled,  controlled,  measured,  and  set  to  work.  Also 
explaining  the  operation  of  Electrical  Apparatus.  Illustrated.  8vo., 
cloth,  140  pp $1 .00 


LIST  OF  WORKS  ON  ELECTRICAL  SCIENCE.  3 

CLARK,  D.  K.     Tramways,  Their  Construction  and  Working.     Second  Edition. 
Illustrated.     8vo.,  cloth,  758  pp $9 .00 

COOPER,  W.  R.  Primary  Batteries:  their  Theory,  Construction,  and  Use.  131 
Illustrations.  8vo.,  cloth,  324  pp Net,  $4 .00 

The  Electrician  Primers.  Being  a  series  of  helpful  primers  on  electrical 
subjects,  for  the  use  of  students,  pupils,  artisans,  and  general  readers. 
Second  Edition.  Illustrated.  Three  volumes  in  one.  8 vo.,  cloth,  Net,  $5.00 

Vol.  I.— Theory  Net,  $2  .00 

Vol.  II.— Electric  Traction,  Lighting  and  Power Net,  $3.00 

Vol.  III.— Telegraphy,  Telephony,  etc Net,  $2 .00 

CROCKER,  F.  B.     Electric  Lighting.     A  Practical  Exposition  of  the  Art  for  the 

use  of  Electricians,  Students,  and  others  interested  in  the  Installation  or 

Operation  of  Electric-Lighting  Plants. 
y0l    i — The  Generating  Plant.     Sixth  Edition,  entirely  revised.      Illustrated, 

8vo.,  cloth,  482  pp $3.00 

Vol.  II. — Distributing  System  and  Lamps.     Sixth  Edition.     Illustrated.     8vo.r 

cloth,  505  pp $3.00 

and  ARENDT,  M.     Electric  Motors:    Their  Action,  Control,  and  Application, 

Illustrated.    8vo.,  cloth In  Press 

and  WHEELER,  S.  S.     The  Management  of  Electrical  Machinery.     Being  a 

thoroughly  revised  and  rewritten  edition  of  the  authors'  "  Practical  Manage- 
ment of  Dynamos  and  Motors."  Seventh  Edition.  Illustrated.  16mo., 
cloth,  232  pp i Net,  $1 .00 

CUSHING,  H.  C.,  Jr.  Standard  Wiring  for  Electric  Light  and  Power.  Illustrated. 
16mo.,  leather,  156  pp $1 .00 

DAVIES,  F.  H.  Electric  Power  and  Traction.  Illustrated.  8vo.,  cloth,  293  pp. 
(Van  Nostrand's  Westminster  Series.) Net,  $2 . 00 

DIBDIN,  W.  J.  Public  Lighting  by  Gas  and  Electricity.  With  many  Tables, 
Figures,  and  Diagrams.  Illustrated.  8vo.,  cloth,  537  pp Net,  $8.00 

DINGER,  Lieut.  H.  C.  Handbook  for  the  Care  and  Operation  of  Naval  Machinery. 
Second  Edition.  Illustrated.  16mo.,  cloth,  302  pp Net,  $2 .00 

DYNAMIC  ELECTRICITY ;  Its  Modern  Use  and  Measurement,  chiefly  in  its  appli- 
catiwii  to  Electric  Lighting  and  Telegraphy,  including:  1.  Some  Points  in 
Electric  Lighting,  by  Dr.  John  Hopkinson.  2.  On  the  Treatment  of  Elec- 
tricity for  Commercial  Purposes,  by  J.  N.  Shoolbred.  3.  Electric-Light 
Arithmetic,  by  R.  E.  Day,  M.E.  Fourth  Edition.  Illustrated.  16mo., 
boards,  166  pp.  (No.  71  Van  Nostrand's  Science  Series.) 50  cents 

EDGCUMBE,  K.  Industrial  Electrical  Measuring  Instruments.  Illustrated.  8vo., 
cloth,  227  pp Net,  $2.50 

ERSKINE-MURRAY,  J.  A  Handbook  of  Wireless  Telegraphy:  Its  Theory  and 
Practice.  For  the  use  of  electrical  engineers,  students,  and  operators. 
Illustrated.  8vo.,  cloth,  320  pp Net,  $3.50 


4  LIST  OF  WORKS  ON  ELECTRICAL  SCIENCE 

EWING,  J.  A.  Magnetic  Induction  in  Iron  and  other  Metals.  Third  Edition,  revised. 
Illustrated.  8vo.,  cloth,  393  pp Net  $4.00 

FISHER,  H.  K.  C.,  and  DARBY,  W.  C.  Students'  Guide  to  Submarine  Cable  Test- 
ing. Third  Edition,  enlarged.  Illus.  8vo.,  cloth,  326  pp Net,  $3.50 

FLEMING,  J.  A.     The  Alternate-Current  Transformer  in  Theory  and  Practice. 
Vol.  I.:    The  Induction  of  Electric  Currents.     Fifth  Issue.     Illustrated.    8vo., 

cloth,  641  pp Net,  $5.00 

Vol.    II.:    The    Utilization    of    Induced    Currents.     Third   Issue.     Illustrated. 

8vo,  cloth,  587  pp Net,  $5 .00 

Handbook  for  the  Electrical  Laboratory  and  Testing  Room.     Two   Volumes. 

Illustrated.     8vo.,  cloth,  1160  pp.     Each  vol Net,  $5.00 

FOSTER,  H.  A.  With  the  Collaboration  of  Eminent  Specialists.  Electrical  Engi- 
neers' Pocket  Book.  A  handbook  of  useful  data  for  Electricians  and 
Electrical  Engineers.  With  innumerable  Tables,  Diagrams,  and  Figures. 
The  most  complete  book  of  its  kind  ever  published,  treating  of  the  latest 
and  best  Practice  in  Electrical  Engineering.  Fifth  Edition,  completely 
revised  and  enlarged.  Fully  Illustrated.  Pocket  Size.  Leather.  Thumb 
Indexed.  1636  pp $5.00 

GANT,  L.  W.  Elements  of  Electric  Traction  for  Motormen  and  Others.  Illustrated 
with  Diagrams.  8vo.,  cloth,  217  pp Net,  $2.50 

GERHARDI,  C.  H.  W.  Electricity  Meters;  their  Construction  and  Management. 
A  practical  manual  for  central  station  engineers,  distribution  engineers 
and  students.  Illustrated.  8vo.,  cloth,  337  pp Net,  $4  .00 

GORE,  GEORGE.  The  Art  of  Electrolytic  Separation  of  Metals  (Theoretical  and 
Practical).  Illustrated.  8yo.,  cloth,  295  pp Net,  $3.50 

GRAY,  J.  Electrical  Influence  Machines:  Their  Historical  Development  and 
Modern  Forms.  With  Instructions  for  making  them.  Second  Edition, 
revised  and  enlarged.  With  105  Figures  and  Diagrams.  12mo.,  cloth, 
296  pp $2.00 

HAMMER,  W.  J.  Radium,  and  Other  Radio-Active  Substances;  Polonium,  Actin- 
ium, and  Thorium.  With  a  consideration  of  Phosphorescent  and  Fluo- 
rescent Substances,  the  properties  and  applications  of  Selenium,  and  the 
treatment  of  disease  by  the  Ultra- Violet  Light.  With  Engravings  and 
Plates.  8vo.,  cloth,  72  pp $1 .00 

HARRISON,  N.  Electric  Wiring  Diagrams  and  Switchboards.  Illustrated.  12mo., 
cloth,  272  pp $1  .£0 

HASKINS,  C.  H.  The  Galvanometer  and  its  Uses.  A  Manual  for  Electricians 
and  Students.  Fifth  Edition,  revised.  Illus.  16mo.,  morocco,  75  pp .  .  $1 . 50 

HAWKINS,  C.  C.,  and  WALLIS,  F.  The  Dynamo:  Its  Theory,  Design,  and  Manu- 
facture. Fourth  Edition,  revised  and  enlarged.  190  Illustrations.  8vo., 
cloth,  925  pp $3 . 00 


LIST  QF  WORKS  ON  ELECTRICAL  SCIENCE.  5 

HAY,  ALFRED.  Principles  of  Alternate-Current  Working.  Second  Edition. 
Illustrated.  12mo.,  cloth,  390  pp $2.00 

Alternating  Currents;  their  theory,  generation,  and  transformation.  Second 
Edition.  1178  Illustrations.  8vo.,  cloth,  319  pp Net  $2.50 

An  Introductory  Course  of  Continuous-Current  Engineering.  Illustrated. 
8vo.,  cloth,  327  pp Net,  $2 . 50 

HEAVISIDE,  0.  Electromagnetic  Theory.  Two  Volumes  with  Many  Diagrams-. 
8vo.,  cloth,  1006  pp.  Each  vol Net,  $5.00 

HEDGES,  K.  Modern  Lightning  Conductors.  An  illustrated  Supplement  to  the 
Report  of  the  Research  Committee  of  1905,  with  notes  as  to  methods  of 
protection  and  specifications.  Illustrated.  8vo.,  cloth,  1 19  pp. .  Net,  $3 .00 

HOBART,  H.  M.  Heavy  Electrical  Engineering.  Illustrated.  8vo.,  cloth,  307 
pp In  Press 

HOBBS,  W.  R.  P.  The  Arithmetic  of  Electrical  Measurements.  With  numerous 
examples,  fully  worked.  Twelfth  Edition.  12mo.,  cloth,  126  pp..  .50  cents 

ROMANS,  J.  E.  A  B  C  of  the  Telephone.  With  269  Illustrations.  12mo., 
cloth,  352  pp $1 .00 

HOPKINS,  N.  M.  Experimental  Electrochemistry,  Theoretically  and  Practically 
Treated.  Profusely  illustrated  with  130  new  drawings,  diagrams,  and 
photographs,  accompanied  by  a  Bibliography.  Illustrated.  8vo.,  cloth, 
298  pp Net,  $3.00 

HOUSTON,  EDWIN  J.     A  Dictionary  of  Electrical  Words,  Terms,  and  Phrases. 

Fourth  Edition,   rewritten  and  greatly  enlarged.      582   Illustrations.      4to., 
cloth Net,  $7.00 

A  Pocket  Dictionary  of  Electrical  Words,  Terms,  and  Phrases.  12mo.,  cloth, 
950  pp Net,  $2.50 

HUTCHINSON,  R.  W.,  Jr.  Long-Distance  Electric  Power  Transmission:  Being 
a  Treatise  on  the  Hydro-Electric  Generation  of  Energy;  Its  Transformation, 
Transmission,  and  Distribution.  Second  Edition.  Illustrated.  12mo., 
cloth,  350  pp Net,  $3 .00 

—  and  IHLSENG,  M.  C.  Electricity  in  Mining.  Being  a  theoretical  and  prac- 
tical treatise  on  the  construction,  operation,  and  maintenance  of  electrical 
mining  machinery.  Illustrated.  12mo.,  cloth In  Press 

INCANDESCENT  ELECTRIC  LIGHTING.  A  Practical  Description  of  the  Edison 
System,  by  H.  Latimer.  To  which  is  added:  The  Design  and  Operation  of 
Incandescent  Stations,  by  C.  J.  Field;  A  Description  of  the  Edison  Electro- 
lyte Meter,  by  A.  E.  Kennelly;  and  a  Paper  on  the  Maximum  Efficiency  of 
Incandescent  Lamps,  by  T.  W.  Howell.  Fifth  Edition.  Illustrated. 
16mo.,  cloth,  140  pp.  (No.  57  Van  Nostrand's  Science  Series.) 50  cents 

INDUCTION  COILS:  How  Made  and  How  Used.  Eleventh  Edition.  Illustrated. 
16mo.,  cloth,  123  pp.  (No.  53  Van  Nostrand's  Science  Series.).  .  .50  cents 


6  LIST  OF  WORKS  ON  ELECTRICAL  SCIENCE, 

JEHL,  FRANCIS,  Member  A.I.E.E.  The  Manufacture  of  Carbons  for  Electric 
Lighting  and  other  purposes.  Illustrated  with  numerous  Diagrams,  Tables, 
and  Folding  Plates.  8vo.,  cloth,  232  pp Net,  $4.00 

JONES,  HARRY  C.  The  Electrical  Nature  of  Matter  and  Radioactivity.  12mo., 
cloth,  212  pp $2.00 

KAPP,  GISBERT.  Electric  Transmission  of  Energy  and  its  Transformation, 
Subdivision,  and  Distribution.  A  Practical  Handbook.  Fourth  Edition, 
thoroughly  revised.  Illustrated.  12mo.,  cloth,  445  pp $3.50 

Alternate-Current  Machinery.     Illustrated.     16mo.,   cloth,   190  pp.     (No.   96 
Van  Nostrand's  Science  Series.) 50  cents 

Dynamos,     Alternators,     and    Transformers.     Illustrated.     8vo.,    cloth,    507 
pp $4.00 

KELSEY,  W.  R.  Continuous-Current  Dynamos  and  Motors,  and  their  Control; 
being  a  series  of  articles  reprinted  from  the  "Practical  Engineer,"  and  com- 
pleted by  W  R.  Kelsey,  B.Sc.  With  Tables,  Figures,  and  Diagrams.  8vo., 
cloth,  439  pp . . .$2.50 

KEMPE,  H.  R.  A  Handbook  of  Electrical  Testing.  Seventh  Edition, 
revised  and  enlarged.  285  Illustrations.  8vo.,  cloth,  706  pp.  .  .  .Net,  $6.00 

KENNEDY,  R.  Modern  Engines  and  Power  Generators.  Illustrated.  4to., 
cloth,  5  vols.  Each $3.50 

Electrical   Installations   of   Electric    Light,   Power,  and   Traction    Machinery. 
Illustrated.     Svo.,  cloth,  5  vols.     Each $3 . 50 

KENNELLY,  A.  E.  Theoretical  Elements  of  Electro-Dynamic  Machinery.  Vol  I. 
Illustrated.  8vo.,  cloth,  90  pp $1 .50 

KERSHAW,  J.  B.  C.  The  Electric  Furnace  in  Iron  and  Steel  Production.  Illus- 
trated. 8vo.,  cloth,  74  pp Net,  $1 .50 

Electrometallurgy.     Illustrated.     8vo.,  cloth,  303  pp.     (Van  Nostrand's  West- 
minster Series.) Net,  $2 . 00 

KINZBRUNNER,  C.  Continuous-Current  Armatures;  their  Winding  and  Con- 
struction. 79  Illustrations.  8vo.,  cloth,  80  pp Net,  $1 .50 

Alternate-Current  Windings;    their  Theory  and  Construction.     89  Illustrations. 
8vo.,  cloth,  80  pp Net,  $1 . 50 

KOESTER,  F.  Steam-Electric  Power  Plants.  A  practical  treatise  on  the  design 
of  central  light  and  power  stations  and  their  economical  construction  and 
operation.  Fully  Illustrated.  4to.,  cloth,  455  pp Net,  $5.00 

LARNER,  E.  T.  The  Principles  of  Alternating  Currents  for  Students  of  Electrical 
Engineering.  Illustrated  with  Diagrams.  12mo.,  cloth,  144  pp. Net,  $1.50 

LEMSTROM,  S.     Electricity  in  Agriculture  and  Horticulture.     Illustrated.     8vo., 
.      cloth Net,  $1 .50 


LIST  OF  WORKS  ON  ELECTRICAL  SCIENCE.  7 

LIVERMORE,  V.  P.,  and  WILLIAMS,  J.  How  to  Become  a  Competent  Motorman: 
Being  a  practical  treatise  on  the  proper  method  of  operating  a  street-railway 
motor-car;  also  giving  details  how  to  overcome  certain  defects.  Second 
Edition.  Illustrated.  16mo.,  cloth,  247  pp Net,  $1 .00 

LOCKWOOD,  T.  D.  Electricity,  Magnetism,  and  Electro-Telegraphy.  A  Prac- 
tical Guide  and  Handbook  of  General  Information  for  Electrical  Students, 
Operators,  and  Inspectors.  Fourth  Edition.  Illustrated.  8vo.,  cloth, 
374  pp $2 .50 

LODGE,  OLIVER  J.  Signalling  Across  Space  Without  Wires:  Being  a  description 
of  the  work  of  Hertz  and  his  successors.  Third  Edition.  Illustrated.  8vo., 
cloth Net,  $2.00 

LORING,  A.  E.  A  Handbook  of  the  Electro-Magnetic  Telegraph.  Fourth  Edition, 
revised.  Illustrated.  16mo.,  cloth,  116  pp.  (No.  39  Van  Nostrand's 
Science  Series.) 50  cents 

LUPTON,  A.  PARR,  G.  D.  A.,  and  PERKIN,  H.  Electricity  Applied  to  Mining. 
Second  Edition.  With  Tables,  Diagrams,  and  Folding  Plates.  8vo..  cloth, 
320  pp Net,  $4.50 

MAILLOUX,     C.     0.     Electric     Traction     Machinery.     Illustrated.     8vo.,     cloth. 

In  Press 

MANSFIELD,  A.  N.  Electromagnets:  Their  Design  and  Construction.  Second 
Edition.  Illustrated.  ICmo.,  cloth,  155  pp.  (Van  Nostrand's  Science 
Series  No.  64.) 50  cents 

MASSIE,  W.  W.,  and  UNDERBILL,  C.  R.  Wireless  Telegraphy  and  Telephony 
Popularly  Explained.  Illustrated.  12mo.,  cloth,  82  pp Net,  $1 .00 

MAURICE,  W.  Electrical  Blasting  Apparatus  and  Explosives,  with  special  refer- 
ence to  colliery  practice.  Illustrated.  8vo.,  cloth,  167  pp Net,  $3.50 

MAVER,  WM.,  Jr.  American  Telegraphy  and  Encyclopedia  of  the  Telegraph  Sys- 
tems, Apparatus,  Operations.  Fifth  Edition,  revised.  450  Illustrations. 
8vo.,  cloth,  656  pp Net,  $5.00 

MONCKTON,  C.  C.  F.     Radio  Telegraphy.     173  Illustrations.     8vo.,  cloth,  272  pp. 

(Van  Nostrand's  Westminster  Series.) Net,  $2 .00 

MUNRO,  J.,  and  JAMIESON,  A.  A  Pocket-Book  of  Electrical  Rules  and  Tables 
for  the  Use  of  Electricians,  Engineers,  and  Electrometallurgists.  Eighteenth 
Revised  Edition.  32mo.,  leather,  735  pp $2 . 50 

NIPHER,  FRANCIS  E.  Theory  of  Magnetic  Measurements.  With  an  Appendix 
on  the  Method  of  Least  Squares.  Illustrated.  12mo.,  cloth,  94  pp.  .  .$1 .00 

NOLL,  AUGUSTUS.  How  to  Wire  Buildings.  A  Manual  of  the  Art  of  Interior 
Wiring.  Fourth  Edition.  Illustrated.  12mo.,  cloth,  165  pp $1 .50 

OHM,  G.  S.  The  Galvanic  Circuit  Investigated  Mathematically.  Berlin,  1827. 
Translated  by  William  Francis.  With  Preface  and  Notes  by  the  Editor, 
Thos.  D.  Lockwood.  Second  Edition.  Illustrated.  16mo.,  cloth,  269  pp. 
(No.  102  Van  Nostrand's  Science  Series.) • 50  cents 


8  LIST  OF  WORKS  ON  ELECTRICAL  SCIENCE. 

OUDIN,  MAURICE  A.  Standard  Polyphase  Apparatus  and  Systems.  Illustrated 
with  many  Photo-reproductions,  Diagrams,  and  Tables.  Fifth  Edition,  revised. 
8vo.,  cloth,  369  pp Net,  $3 .00 

PALAZ,  A.  Treatise  on  Industrial  Photometry.  Specially  applied  to  Electric 
Lighting.  Translated  from  the  French  by  G.  W.  Patterson,  Jr.,  Assistant 
Professor  of  Physics  in  the  University  of  Michigan,  and  M.  R.  Patterson, 
B.A.  Second  Edition.  Fully  Illustrated.  8vp.,  cloth,  324  pp $4.00 

PARR,  G.  D.  A.  Electrical  Engineering  Measuring  Instruments  for  Commercial 
and  Laboratory  Purposes.  With  370  Diagrams  and  Engravings.  8vo., 
cloth,  328  pp Net,  $3 . 50 

PARSHALL,  H.  F.,  and  HOBART,  H.  M.  Armature  Windings  of  Electric  Machines. 
Third  Edition.  With  140  full-page  Plates,  65  Tables,  and  165  pages  of 

descriptive  letter-press.     4to.,  cloth,  300  pp $7.50 

Electric   Railway   Engineering.     With   437   Figures   and   Diagrams   and   many 
Tables.     4to.,  cloth,  475 pp Net,  $10.00 

Electric  Machine  Design.     Being  a  revised  and  enlarged  edition  of  "  Electric 
Generators."     648  Illustrations.     4to,  half  morocco,  601  pp.  . .  .Net,  $12.50 

PERRINE,  F.  A.  C.  Conductors  for  Electrical  Distribution:  Their  Manufacture 
and  Materials,  the  Calculation  of  Circuits,  Pole-Line  Construction,  Under- 
ground  Working,  and  other  Uses.  Second  Edition.  Illustrated.  8vo., 
cloth,  287  pp Net,  $3.50 

POOLE,  C.  P.  Wiring  Handbook  with  complete  Labor-saving  Tables  and  Digest 
of  Underwriters'  Rules.  Illustrated.  12mo.,  leather,  85  pp Net,  $1 .00 

POPE,  F.  L.  Modern  Practice  of  the  Electric  Telegraph.  A  Handbook  for  Elec- 
tricians and  Operators.  Seventeenth  Edition.  Illustrated.  8vo.,  cloth, 
234  pp $1 .50 

RAPHAEL,  F.  C.  Localization  of  Faults  in  Electric-Light  Mains.  Second  Edition, 
revised.  Illustrated.  8vo.,  cloth,  205  pp Net,  $3 .00 

RAYMOND,  E.  B.  Alternating-Current  Engineering,  Practically  Treated.  Third 
Edition,  revised.  With  many  Figures  and  Diagrams.  8vo.,  cloth,  244  pp., 

Net,  $2.50 

ROBERTS,  J.  Laboratory  Work  in  Electrical  Engineering— Preliminary  Grade. 
A  series  of  laboratory  experiments  for  first-  and  second-year  students  in 
electrical  engineering.  Illustrated  with  many  Diagrams.  8vo.,  cloth, 
218  pp Net,  $2 .00 

ROLLINS,  W.  Notes  on  X-Light.  Printed  on  deckle  edge  Japan  paper.  400 
pp.  of  text,  152  full-page  plates.  8vo.,  cloth Net,  $7.50 

RUHMER,  ERNST.  Wireless  Telephony  in  Theory  and  Practice.  Translated 
from  the  German  by  James  Erskine-Murray.  Illustrated.  8vo.,  cloth, 
224  pp Net,  $3.50 

RUSSELL,  A.  The  Theory  of  Electric  Cables  and  Networks.  71  Illustrations. 
Svo.,  cloth,  275  pp Net,  $3 . 00 


LIST  OF  WORKS  ON  ELECTRICAL  SCIENCE.  9 

SALOMONS,  DAVID.  Electric-Light  Installations.  A  Practical  Handbook.  Illus- 
trated. 12mo.,  cloth. 

Vol  I.:    Management  of  Accumulators.     Ninth  Edition.     178  pp $2.50 

Vol.  II. :    Apparatus.     Seventh  Edition.      318  pp $2 . 25 

Vol.  III. :    Application.     Seventh  Edition.     234  pp $1 . 50 

SCHELLEN,  H.  Magneto-Electric  and  Dynamo-Electric  Machines.  Their  Con- 
struction and  Practical  Application  to  Electric  Lighting  and  the  Trans- 
mission of  Power.  Translated  from  the  Third  German  Edition  by  N.  S. 
Keith  and  Percy  Neymann,  Ph.D.  With  very  large  Additions  and  Notes 
relating  to  American  Machines,  by  N.  S.  Keith.  Vol.  I.  With  353  Illus- 
trations. Third  Edition.  8vo.,  cloth,  518  pp $5.00 

SEVER,  G.  F.  Electrical  Engineering  Experiments  and  Tests  on  Direct-Current 
Machinery.  Second  Edition,  enlarged.  With  Diagrams  and  Figures.  8vo., 
pamphlet,  75  pp Net,  $1 .00 

and  TOWNSEND,  F.     Laboratory  and  Factory  Tests  in  Electrical  Engineering. 

Second  Edition.     Illustrated.     8vo.,  cloth,  269  pp Net,  $2 .50 

SEWALL,    C.    H.     Wireless   Telegraphy.     With    Diagrams    and    Figures.     Second 

Edition,  corrected.     Illustrated.     8vo.,  cloth,  229  pp Net,  $2.00 

Lessons  in  Telegraphy.     Illustrated.     12mo.,  cloth,  104  pp Net,  $1 .00 

T.     Elements  of  Electrical  Engineering.     Third  Edition,  revised.     Illustrated. 

8vo.,  cloth,  444  pp $3.00 

The  Construction  of  Dynamos  (Alternating  and  Direct  Current).  A  Text- 
book for  students,  engineering  contractors,  and  electricians-in-charge. 
Illustrated.  8vo.,  cloth,  316  pp $3 . 00 

SHAW,  P.  E.  A  First- Year  Course  of  Practical  Magnetism  and  Electricity.  Spe- 
cially adapted  to  the  wants  of  technical  students.  Illustrated.  8vo., 
cloth,  66  pp.  interleaved  for  note  taking Net,  $1 .00 

SHELDON,   S.,  and  MASON,  H.     Dynamo-Electric  Machinery:    Its  Construction, 

Design,  and  Operation. 

Vol.  I.:  Direct-Current  Machines.  Seventh  Edition,  revised.  Illustrated. 
8vo.,  cloth,  281  pp :  .Net,  $2.50 

and  HAUSMANN,  E.    Alternating-Current  Machines :  Being  the  second  volume 

of  "  Dynamo-Electric  Machinery;  its  Construction,  Design,  and  Opera- 
tion." With  many  Diagrams  and  Figures.  (Binding  uniform  with  Vol- 
ume I.)  Seventh  Edition,  rewritten.  8vo.,  cloth,  353  pp Net,  $2 . 50 

SLOANE,  T.  O'CONOR.  Standard  Electrical  Dictionary.  300  Illustrations.  12mo., 
cloth,  682  pp $3.00 

Elementary  Electrical  Calculations.  How  Made  and  Applied.  Illustrated. 
8vo.,  cloth,  300  pp In  Press 

SNELL,  ALBION  T.  Electric  Motive  Power.  The  Transmission  and  Distribution 
of  Electric  Power  by  Continuous  and  Alternating  Currents.  With  a  Section 
on  the  Applications  of  Electricity  to  Mining  Work.  Second  Edition. 
Illustrated.  8vo.,  cloth,  411  pp Net,  $4.00 


10  LIST  OF  WORKS  ON  ELECTRICAL  SCIENCE. 

SODDY,  F.  Radio-Activity ;  an  Elementary  Treatise  from  the  Standpoint  of  the 
Disintegration  Theory.  Fully  Illustrated.  8vo.,  cloth,  214  pp.  .Net,  $3.00 

SOLOMON,  MAURICE.  Electric  Lamps.  Illustrated.  8vo.,  cloth.  (Van  Nos- 
trand's  Westminster  Series.) Net,  $2 . 00 

STEWART,  A.  Modern  Polyphase  Machinery.  Illustrated.  12mo.,  cloth,  296 
pp Net,  $2.00 

SWINBURNE,  JAS.,  and  WORDINGHAM,  C.  H.  The  Measurement  of  Electric 
Currents.  Electrical  Measuring  Instruments.  Meters  for  Electrical  Energy. 
Edited,  with  Preface,  by  T.  Commerford  Martin.  Folding  Plate  and  Numer- 
ous Illustrations.  16mo.,  cloth,  241  pp.  (No.  109  Van  Nostrand's  Science 
Series.) 50  cents 

SWOOPE,  C.  WALTON.  Lessons  in  Practical  Electricity:  Principles,  Experi- 
ments, and  Arithmetical  Problems.  An  Elementary  Text-book.  With 
numerous  Tables,  Formulae,  and  two  large  Instruction  Plates.  Ninth 
Edition.  Illustrated.  8vo.,  cloth,  462  pp Net,  $2 .00 

THOM,  C.,  and  JONES,  W.  H.  Telegraphic  Connections,  embracing  recent  methods 
in  Quadruplex  Telegraphy.  20  Colored  Plates.  8vo.,  cloth,  59  pp.  .$1 .50 

THOMPSON,  S.  P.,  Prof.  Dynamo-Electric  Machinery.  With  an  Introduction 
and  Notes  by  Frank  L.  Pope  and  H.  R.  Butler.  Fully  Illustrated.  16mo., 
cloth,  214  pp.  (No.  66  Van  Nostrand's  Science  Series.).  .  . 50  cents 

Recent  Progress  in  Dynamo-Electric  Machines.  Being  a  Supplement  to 
"Dynamo-Electric  Machinery."  Illustrated.  16mo.,  cloth,  113  pp.  (No. 
75  Van  Nostrand's  Science  Series.) 50  cents 

TOWNSEND,  FITZHUGH.  Alternating  Current  Engineering.  Illustrated.  8vo., 
paper,  32  pp Net,  75  cents 

UNDERBILL,  C.  R.  The  Electromagnet:  Being  a  new  and  revised  edition  of 
"The  Electromagnet,"  by  Townsend  Walcott,  A.  E.  Kennelly,  and  Richard 
Varley.  With  Tables  and  Numerous  Figures  and  Diagrams.  12mo., 
cloth New  Revised  Edition  in  Press 

URQUHART,  J.  W.  Dynamo  Construction.  A  Practical  Handbook  for  the  use 
of  Engineer  Constructors  and  Electricians  in  Charge.  Illustrated.  12mo., 
cloth $3.00 

Electric  Ship-Lighting.  A  Handbook  on  the  Practical  Fitting  and  Running  of 
Ship's  Electrical  Plant,  for  the  use  of  Ship  Owners  and  Builders,  Marine 
Electricians,  and  Sea-going  Engineers  in  Charge.  88  Illustrations.  12mo., 
cloth,  308  pp $3 .00 

Electric-Light  Fitting.  A  Handbook  for  Working  Electrical  Engineers,  em- 
bodying Practical  Notes  on  Installation  Management.  Second  Edition. 
With  numerous  Illustrations.  12mo.,  cloth $2 .00 

Electroplating.     Fifth  Edition.     Illustrated.     12mo.,  cloth,  230  pp $2.00 

Electrotyping.     Illustrated.     12mo.,  cloth,  228  pp $2 . 00 


LIST  OF   WORKS  ON  ELECTRICAL  SCIENCE.  11 

WADE,  E.  J.  Secondary  Batteries:  Their  Theory,  Construction,  and  Use.  With 
innumerable  Diagrams  and  Figures.  8vo.,  cloth New  Edition  in  Press 

WALKER,  FREDERICK.  Practical  Dynamo-Building  for  Amateurs.  How  to 
Wind  for  any  Output.  Third  Edition.  Illustrated.  16mo.,  cloth,  104  pp. 
(No.  98  Van  Nostrand's  Science  Series.) 50  cents 

— —  SYDNEY  F.  Electricity  in  Homes  and  Workshops.  A  Practical  Treatise  on 
Auxiliary  Electrical  Apparatus.  Fourth  Edition.  Illustrated.  12mo., 
cloth,  358  pp.  . $2 .00 

Electricity  in  Mining.     Illustrated.     8vo.,  cloth,  385  pp $3 .50 

WALLING,  B.  T.,  Lieut.-Com.  U.S.N.,  and  MARTIN,  JULIUS.     Electrical  Installa- 
tions of  the  United  States  Navy.     With  many  Diagrams  and  Engravings. 
'  8vo.,  cloth,  648  pp . $6.00 

WALMSLEY,  R.  M.  Electricity  in  the  Service  of  Man.  A  Popular  and  Practical 
Treatise  on  the  Application  of  Electricity  in  Modern  Life.  Illustrated. 
8vo.,  cloth,  1208  pp Net,  $4.50 

WATT,    ALEXANDER.     Electroplating    and    Refining    of    Metals.     New  'Edition, 

rewritten  by' Arnold  Philip.     Illustrated.      8vo.,  cloth,  677  pp.  .Net,  $4.50 

Electro-Metallurgy.     Fifteenth    Edition.     Illustrated.     12mo.,    cloth,    225    pp., 

$1.00 

WEBB,  H.  L.  A  Practical  Guide  to  the  Testing  of  Insulated  Wires  and  Cables. 
Fifth  Edition.  Illustrated.  12mo.,  cloth,  118  pp $1 .00 

WEEKS,   R.   W.     The   Design   of  Alternate-Current  Transformer.      New  Edition 

in  Press 

WEYMOUTH,  F.  MARTEN.  Drum  Armatures  and  Commutators.  (Theory  and 
Practice.)  A  complete  treatise  on  the  theory  and  construction  of  drum- 
winding,  and  of  commutators  for  closed-coil  armatures,  together  with  a  full 
resume  of  some  of  the  principal  points  involved  in  their  design,  and  an 
exposition  of  armature  reactions  and  sparking.  Illustrated.  8vo.,  cloth, 
295  pp .Net,  $3.00 

WILKINSON,  H.  D.  Submarine  Cable-Laying,  Repairing,  and  Testing.  New  Edition. 
Illustrated.  8vo.,  cloth In  Press 

YOUNG,  J.  ELTON.  Electrical  Testing  for  Telegraph  Engineers.  Illustrated. 
8vo.,  cloth,  264  pp Net,  $4.00 

A  112=page  Catalog  of  Books  on  Electricity,  classified  by 
subjects,  will  be  furnished  gratis,  postage  prepaid,  on 
application. 


THE  NEW  FOSTER 


Fifth  Edition,  Completely  Revised  and 
Enlarged,  with  Four-fifths  of  Old  Matter 
Replaced  by  New,  Up-to-date  Material. 
Pocket  size,  flexible  leather,  elaborately 
illustrated,  with  an  extensive  index,  1636 
pp.,  Thumb  Index,  etc.  Price,  $5.00. 

ELECTRICAL 

ENGINEER'S 

POCKETBOOK 

The  Most  Complete  Book  of  Its  Kind  Ever  Published, 

Treating  of  the  Latest  and  Best  Practice 

in  Electrical  Engineering 

By  HORATIO  A.  FOSTER 

Member  Am.  Inst.  E.  E.,  Member  Am.  Soc.  M.  E. 

With   the   Collaboration  of   Eminent  Specialists 


Symbols,  Units,  Instruments 

Measurements 

Magnetic  Properties  of  Iron 

Electro-Magnets 

Properties  of  Conductors 

Relations  and  Dimensions  of 
Conductors 

Jnderground  Conduit  Con- 
struction 

Standard  Symbols 

Zable  Testing 

3ynamos  and  Motors 

Tests  of  Dynamos  and  Motors 


CONTENTS 

The  Static  Transformer 
Standardization  Rules 
Illuminating  Engineering 
Electric  Lighting  (Arc) 

(Incandescent) 
Electric  Street  Railways 
Electrolysis 

Transmission  of  Power 
Storage  Batteries 
Switchboards 
Lightning  Arresters 
Electricity  Meters 
Wireless  Telegraphy 
Telegraphy 


Telephony 

Electricity  in  the  U.  S.  Army 

Electricity  in  the  U.  S.  Navy 

Resonance 

Electric  Automobiles 

Electro-chemistry  and  Electro- 
metallurgy 

X-Rays 

Electric  Heating,  Cooking  and 
Welding 

Lightning  Conductors 

Mechanical  Section 

Index 


D.  VAN  NO5TRAND   COHPANY, 

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23  MURRAY  AND  27  WARPEN  STREETS.  NEW 


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